Tunable polaritonic topologies generated by non-local photonic modes

Topology supplies a foundational framework for understanding a variety of pure phenomena^1,2,3. Amongst its key manifestations are topological defects, which can’t be eliminated or reworked with out basically altering the system’s configuration, intrinsically stopping their decay. The skyrmion⁴ is a main instance, consisting of a three-dimensional (3D) vector subject mapped onto a two-dimensional (2D) aircraft. It’s usually described as a vector subject encoding distinct mappings on a 3D unit sphere in order-parameter house, capturing the winding and twisting of the sector. A skyrmion is characterised by totally protecting the unit sphere such that each one doable orientations of the vector subject are represented. In condensed matter and solid-state physics, skyrmions seem in techniques starting from magnetic supplies^5,6 to superconductors^7,8, superfluids⁹ and liquid crystals^10,11,12.

Just lately, topological defects have been prolonged to photonics, the place skyrmions had been noticed by way of managed interference of free-space waves^13,14,15,16 and floor plasmon polaritons^17,18,19. These photonic skyrmions exhibit deeply subwavelength options²⁰ and inherent topological robustness towards materials defects and environmental perturbations^21,22,23, highlighting their potential for optical computing, metrology and twistronics²⁴. Additionally they possess non-trivial options similar to topological area partitions, tunable by way of the ratio of in-plane to out-of-plane momentum. This allows transitions from bubble-type skyrmions with sharp area partitions to Néel-type skyrmions with smeared area partitions^17,25. Following their preliminary realization in plasmonics, latest demonstrations embody free-space skyrmions^13,26, skyrmion luggage²⁴ and varied polaritonic topologies, similar to optical meron lattices²⁷, and deeply subwavelength optical vortices carrying orbital angular momentum. The latter had been achieved by interfering floor phonon polaritons in isotropic polar supplies²⁸ (ε_xx = ε_yy = ε_zz) or hyperbolic phonon polaritons (HPhPs) in anisotropic polar supplies^29,30 (ε_xx = ε_yy ≠ ε_zz), with functions in structured thermal emission³¹.

Nevertheless, present approaches for producing polaritonic subject skyrmions^14,25,32 depend on non-reconfigurable, wavelength-dependent constructions similar to gratings or phase-correcting offsets, or on structured gentle similar to radial polarization to launch and intervene floor waves (Fig. 1a, left). These constraints restrict same-structure topological tunability, hindering integration into optical computing platforms requiring broadband reconfigurability. This contrasts with free-space skyrmions, the place tunability has been demonstrated¹³. As well as, excitation usually requires round or radial polarization³³, necessitating further optical components similar to waveplates and rising experimental complexity.

On this work, we introduce structured polaritonic topologies generated by non-local photonic resonances^34,35, enabling skyrmion formation with out phase-correcting offsets and utilizing linearly polarized gentle (Fig. 1a, proper). We understand this by utilizing arrays of excessive refractive-index dielectric hexagonal resonators (Fig. 1b) on a clear CaF₂ substrate supporting quasi-bound states within the continuum (qBICs)^36,37 below linear polarization. These resonances come up from engineered in-plane asymmetry inside every unit cell, permitting management over their linewidths³⁶. Crucially, symmetry-protected qBICs require prolonged periodic arrays relatively than remoted resonators^38,39. This enables a number of resonators to be pushed concurrently by the identical excitation, with topology ruled by resonator geometry. Because of this, optical skyrmions will be scaled from single constructions to photonic chips, enabling large-area metasurfaces with a number of encoded skyrmion lattices.

On the premise of this precept, we experimentally show the era and reconfigurability of qBIC-driven polaritonic topologies by way of interference of HPhPs in hexagonal boron nitride (hBN) skinny movies. Utilizing scattering-type scanning near-field optical microscopy (s-SNOM), we resolve amplitude and section of deeply subwavelength photonic skyrmion lattices induced by the non-local qBIC. By tuning the excitation frequency throughout the Reststrahlen (RS) band of hBN, we dynamically management the diameter D_hex of particular person skyrmions with out modifying the metasurface geometry. Our platform supplies a route in the direction of frequency-encoded topological states as reconfigurable constructing blocks for next-generation quantum photonic platforms.

Non-local mode formation for the era of polaritonic skyrmions

The non-local photonic mode that emerges from our construction depends on sturdy mutual subject interactions between particular person resonators³⁴ that generate out-of-plane electrical fields E_z (Fig. 1c). Importantly, these fields are extremely uniform throughout the resonator floor in each amplitude and section, in distinction to, for instance, these rising from a dipolar resonance (Fig. 1g–i). By protecting every resonator with hBN (Fig. 1d) and tailoring the resonance to lie throughout the in-plane RS band of hBN (Fig. 1e) (the place ε_r,|| 40 on particular person resonators. These modes come up in skinny hBN movies resulting from long-range coulomb interactions and the macroscopic polarization subject that results in a spectral splitting between longitudinal and transverse optical phonons, along with the intrinsic anisotropy of hBN. This anisotropy originates from sturdy in-plane covalent bonding and weaker out-of-plane van der Waals interactions, resulting in sturdy polariton confinement.

A key benefit of our platform is that HPhPs are generated with an identical depth and section at every resonator edge (Supplementary Fig. 1), not like conventional approaches utilizing single resonant constructions^41,42,43. We simulate the out-of-plane electrical subject E_z of the hBN-covered metasurface and observe constructive interference of HPhPs on the resonator centre, forming a lattice of photonic skyrmions (Fig. 1f). This contrasts with native dipolar resonances in single dielectric resonators (Fig. 1g), which produce non-uniform E_z distributions and polarization-dependent depth (Fig. 1h and Supplementary Fig. 2). Thus, native resonances similar to dipolar modes are basically incapable of producing the related topologies (Fig. 1i).

We began our experimental investigation by imaging the all-dielectric metasurface, schematically proven in Fig. 1b. Our design consists of hexagonal amorphous silicon (a-Si) pillars, with every pair laterally offset from each other. A scanning electron microscopy (SEM) picture of the fabricated metasurface is proven in Fig. 2a, and atomic power microscopy (AFM) measurements of single unit cells with hexagonal, round, and sq. resonators are proven in Fig. 2b–d. To spectrally tune the metasurface resonance, we differ the in-plane scaling issue S, which linearly modifies all unit cell dimensions besides the peak of the a-Si pillars. For all experiments, the pitch was set to P_x = 5,250 nm, P_y = 4,725 nm for a scaling issue S = 1 and the peak of the resonators to h_Si = 1,450 nm. For a periodic array of resonators with C₄ symmetry (that’s, no lateral offset), the qBIC manifests as a darkish mode with out radiative loss channels (infinite Q-factor) and can’t be noticed within the far subject. To entry this photonic mode experimentally, the in-plane symmetry inside every unit cell is damaged, opening a radiative loss channel and leading to an observable resonance (Supplementary Fig. 4) with finite radiative Q-factor Q_rad, which will be tuned by offsetting alternating resonator pairs by a distance D_x (Fig. 2b).

To quantify this asymmetry, we outline the asymmetry parameter α as follows:

For all samples fabricated on this work, we select α = 0.045, because it supplies comparatively excessive Q_rad of round 50–100 (Supplementary Fig. 4) whereas sustaining sufficiently broad resonances to experimentally reconfigure the HPhP wavelength, because the linewidth of the qBIC resonance corresponds to the tuning vary of our strategy. Such tunability arises from the sturdy sublinear dispersion of hBN (Supplementary Fig. 4), which allows massive adjustments in polariton wavelength with small adjustments in excitation frequency^28,42. To make sure that the uniform out-of-plane electrical fields will be accessed over a broad vary of HPhP momenta, the metasurface is purposefully designed to help resonances with modest Q-factors.

The native near-fields of the photonic qBIC mode are imaged utilizing transmission-mode s-SNOM with a pointy metallic tip (radius ≈ 50 nm) as an area scatterer (Fig. 2e). The total setup is proven in Supplementary Fig. 5. Because the tip is polarized alongside the shaft, it primarily scatters out-of-plane electrical fields E_z. By focusing a single-wavelength mid-IR beam onto the metasurface at regular incidence and scanning throughout particular person resonators, each the native out-of-plane amplitude |E_z| and section φ_z are extracted by way of pseudo-heterodyne (PsHet) detection⁴³. The measured φ_z for hexagonal resonators is proven in Fig. 2f and agrees properly with simulations (Fig. 1c and Supplementary Fig. 4), exhibiting uniform out-of-plane electrical fields throughout every resonator floor. As well as, edge scans on the metasurface (Supplementary Fig. 6) present that the non-local mode kinds after roughly 6–7 resonators (3–4 unit cells), indicating that only some unit cells are required to generate the qBIC mode within the close to subject, in step with earlier research^38,39.

We show the flexibility and generality of our idea by measuring the φ_z of unit cells with modified resonator shapes, specifically discs (Fig. 2g) and squares (Fig. 2h), along with the hexagonal constructions. The outcomes present excessive uniformity of E_z throughout all geometries, indicating that the strategy is mostly relevant to completely different resonator shapes, offered ample mode quantity is on the market for correct formation of the photonic mode. Additional particulars are given within the Strategies, and a fabrication sketch is proven in Supplementary Fig. 7. For this examine, transmission-mode s-SNOM is most popular over reflection mode, because it allows excitation of the qBIC mode at regular incidence whereas suppressing tip-launched polaritons⁴⁴. This enables the tip to behave as a passive scatterer, detecting near-fields generated by the photonic mode with out perturbing the polaritonic topologies.

To generate qBIC-driven photonic skyrmion lattices localized on particular person resonators, we fabricated dielectric metasurfaces coated with hBN flakes of thickness h_hBN = 50–70 nm. A SiO₂ layer (h_SiO2 = 50 nm) is inserted between a-Si and hBN to boost adhesion and improve polariton lifetimes owing to its decrease refractive index⁴⁵. For all samples, hBN flakes of dimension 50 × 50 to 100 × 100 μm², protecting 10–20 unit cells, had been used. To maximise spatial mode density, we employed a spectral gradient metasurface^46,47 by repeatedly various the in-plane scaling issue S alongside one axis, spatially encoding a variety of resonance wavelengths inside a single array (Supplementary Fig. 8) and lowering the footprint⁴⁶. The spatial encoding was verified by way of large-area near-field scans (Supplementary Fig. 9).

Owing to the excessive tunability of HPhPs with small shifts in excitation frequency in hBN skinny movies^42,48, the broad vary of resonances coated by our metasurface (Fig. 3a) generates HPhPs with drastically completely different wavelengths alongside the gradient (Fig. 3b–d), leading to photonic skyrmion diameters starting from D_hex = 451 nm all the way down to 271 nm. Notice that by definition, D_hex = λ_HPhP. Particular person photonic skyrmions are seen in each the measured section φ_z and amplitude |E_z|, with reducing dimension and rising quantity per resonator at larger excitation wavenumbers and smaller scaling issue S. Our qBIC-driven skyrmions are deeply subwavelength (~λ/25) and are about twice as small as plasmonic skyrmions reported beforehand¹⁷. This sturdy confinement arises from the polariton dispersion, which yields massive in-plane momenta, resulting in an imaginary out-of-plane wavevector and evanescent decay regular to the floor⁴⁹. The HPhP wavelength will be additional diminished by rising the excitation wavenumber or utilizing thinner hBN, probably mixed with sharper tricks to resolve smaller options. Notice that the volumetric HPhPs, that are usually noticed in hBN slabs (>10 nm), would shift in the direction of purely floor modes when approaching the one atomic layer restrict⁵⁰. In precept, this permits an arbitrary variety of photonic skyrmions on a single resonator, permitting localized and optically reprogrammable topological fees (Supplementary Fig. 10). Whereas hBN helps a number of hyperbolic modes at a given excitation wavelength, the dominant mode (m = 0; Supplementary Fig. 3) is primarily detected with s-SNOM²⁹. Though we use α = 0.045 for all fabricated constructions, simulations present that the topology is generated whatever the resonance Q-factor (Supplementary Fig. 11).

Topological reconfigurability of qBIC-driven polaritonic skyrmions

To characterize the topological properties of qBIC-driven photonic skyrmions, we calculated the skyrmion quantity density (SND), which describes the spatial distribution of the sector’s topological traits, and topological winding quantity S_T, which describes what number of occasions the vector subject in a given space σ wraps across the unit sphere. These portions will be written as

the place (hat{mathbf{e}} = (E_{x}, E_{y}, E_{z}) / sqrt{|E_{x}|^2 + |E_{y}|^2 + |E_{z}|^2}) is the normalized electrical subject vector. Hereby, the winding quantity denotes the variety of skyrmions inside any given space σ on the floor of a resonator. The in-plane electrical subject parts will be straight obtained from the out-of-plane electrical subject measured with s-SNOM by Maxwell’s equations, as proven in Supplementary Notice 4 and in earlier works^17,32.

We examine the SND and S_T for the measurement proven in Fig. 3d (D_hex = 271 nm). Every resonator is labelled with a notation of (↑, n) or (↓, n), the place ↑ or ↓ denotes the sector path of E_z for the central skyrmion and (n) distinguishes two resonators of the identical polarity. The calculated SND (Fig. 4a) exhibits a typical Néel-type skyrmion sample with area partitions which can be smeared-out¹⁷, owing to the deeply subwavelength nature of the HPhPs generated in our construction (λ_HPhP ≈ λ₀/25). Normally, Néel-type photonic skyrmions with smeared area partitions are extra readily accessible in supplies that help phonon polaritons, as floor plasmon polaritons with lengthy propagation lengths typically exhibit solely a reasonable discount in wavelength in contrast with the incident gentle⁴⁴. Calculated SNDs for the measurements in Fig. 3b,c are proven in Supplementary Fig. 14.

Inside a cluster of seven adjoining hexagonal cells of the skyrmion lattice (every with diameter D_hex), the winding quantity S_T is near the theoretical worth of ±1 inside every space (Fig. 4a, proper), with the signal relying on the path of E_z. Owing to opposing subject instructions in every resonator pair rising from the non-local qBIC resonance, each S_T values of +1 and −1 are recovered inside every lattice website. This contrasts with photonic skyrmions in non-resonant remoted constructions, the place S_T is solely decided by the section of the incident gentle. Summing all winding numbers for the resonators (↑, 1) and (↓, 1) throughout the seven hexagonal cells yields values of 6.95 and −6.99, respectively, in good settlement with the theoretical worth S_T = ±7, demonstrating the topological robustness of qBIC-driven photonic skyrmion lattices. This stability is additional illustrated by easily various the optical section φ_z (Fig. 4b,c) and calculating the S_T for every worth, the place for (↑, 1) (↑, 2) and (↓, 1) (↓, 2), S_T abruptly switches from +1 to −1 and again to +1, in step with theoretical modelling.

As illustrated in Fig. 1a, our platform circumvents geometrically wavelength-specific offsets for section compensation, relying as an alternative on engineered qBIC resonances mediated by long-range coupling between resonators. We show optical reconfigurability by consecutively imaging the identical resonator whereas tuning the excitation frequency in small steps (∆ω = cm⁻¹), which adjustments λ_HPhP considerably owing to the sturdy dispersion throughout the hBN in-plane RS-band. Our measurements (Fig. 4d) present steady tuning of D_hex throughout the similar resonator from 448 nm to 342 nm, whereas the winding numbers S_T round every central skyrmion stay near the theoretical worth of 1. This tunability window can, in precept, be made arbitrarily massive by broadening the qBIC resonance by way of rising the asymmetry parameter α, enabling the photonic mode to kind over a wider wavelength vary.

Era of arbitrarily structured topologies

Our platform provides an easy path to generate arbitrarily structured optical topologies by variation of the resonator form. As proven in Supplementary Fig. 4, the uniform out-of-plane electrical fields of the non-local qBIC metasurface are preserved when shifting from hexagonal resonators to discs or squares of comparable mode quantity. As with the hexagonal design producing skyrmion lattices, disc and sq. resonators exhibit the identical diploma of E_z uniformity throughout every floor. We anticipate this behaviour to increase to extra complicated resonator shapes resembling a disc, similar to twisted hexagons for the era of skyrmion luggage.

To show the generality of our platform, we experimentally probe polaritonic kπ-twist skyrmions, beforehand solely noticed in singular graphene discs⁵¹, in addition to optical meron lattices, beforehand realized in patterned gold movies²⁷ (Supplementary Fig. 16). The measured okayπ-twist skyrmions are generated in disc resonators (Fig. 5a–d) and encompass concentric rings centred round a single skyrmion with alternating out-of-plane electrical subject instructions. Sq. resonators as an alternative generate optical meron lattices (Fig. 5e–h), with each simulated and measured S_T values proven in Supplementary Fig. 17. Notice that merons are topologically much less secure than skyrmions as they span solely half a unit sphere, leading to S_T values deviating farther from the perfect ±0.5. For big-scale SEM photographs of the fabricated gadgets, see Supplementary Fig. 18. As proven beforehand for the tunable skyrmion lattices in hexagonal resonators, this strategy removes the necessity for wavelength-specific geometries and allows same-structure reconfigurability.