1 Introduction

Low-dimensional materials form an emerging platform for exotic light-matter interactions, ideally suited for various photonic technologies [1,2,3,4,5]. In these materials, light hybridizes with matter to form quasiparticles known as polaritons [6, 7], which feature deeply subwavelength field confinement and broadband responses spanning from terahertz to infrared and visible frequencies. The hyperbolicity of polaritons supported by various natural materials, especially in anisotropic van der Waals (vdW) crystals, is particularly promising for next-generation nanophotonics. The hyperbolicity originates from extreme anisotropy of the atomic interaction between in-plane covalent and out-of-plane vdW bondings [8], enabling unusually large wave momenta associated with giant field confinement and slow group velocities over broad bandwidths. While hyperbolic phenomena have been originally explored in engineered metamaterials, such as metal–insulator-metal multilayers, this approach suffers from inevitable fabrication imperfections and finite granularity, resulting in enhanced loss and nonlocality hindering their widespread applicability [9, 10]. Natural hyperbolic polaritons in polar vdW crystals [11] can overcome these challenges: for instance, hexagonal boron nitride (hBN) conical nanoresonators enable remarkably large quality-factors (~ 283) and light confinement (~ λ/86) [12]. In addition, the dispersion of hyperbolic phonon polaritons (HP2) can be engineered with large flexibility, as shown in hBN [12,13,14,15,16,17] and α-MoO3 [18,19,20,21,22,23], facilitating super-resolution imaging [14, 15], light canalization [17, 20], molecular sensing [24, 25], reconfigurable propagation [26, 27], for a variety of polaritonic applications [11, 28].

So far, the manipulation of hyperbolic phonon polaritons (HP2) has been mainly limited to amplitude control, while adding new degrees of freedom may dramatically expand the impact of polaritonics for information processing and transport. For instance, surface plasmon polaritons (SPPs) have recently been explored to control phase, spin and orbit angular momentum (OAM) as independent degrees of freedom [29,30,31,32,33,34,35]. Based on these principles, a plasmonic vortex has been explored to enhance the channel capacity of on-chip optical communication networks [36]. Plasmonic OAMs can selectively couple to the spin of light excitation, inducing spin–orbit interactions [37, 38] and spin-controlled plasmonic phenomena, such as unidirectional routing [39], plasmonic vortex generation [29, 30, 40, 41] and information detection, opening new opportunities for low-energy information processing and computing. Extending these explorations to the polaritonic regime in the mid-IR range is exciting on various fronts: (1) we can expect significantly reduced loss compared to their plasmonic counterparts; (2) mid-IR frequencies are of particular interest for on-chip communications and sensing, since several biomolecules have fingerprints in this range; (3) endowing phonon polaritons with these new degrees of freedom unlocks new forms of low-energy information transport and processing. Low-dimensional phonon-polariton nanomaterials offer an ideal playground to enable this vision. The lack of exploration of HP2 in the context of phase and angular momentum control may be associated with the fact that most polaritonic materials support bulk polariton modes, in contrast with plasmon polaritons that travel at interfaces, making more difficult the excitation, observation, and direct access to these phenomena.

2 Results and discussion

In this paper, we tackle and address these challenges by coupling HP2s to tailored nanostructures, leveraging refined theoretical design and observing the response via real-space near-field nanoimaging. Specifically, we demonstrate infrared nanoscale HP2 vortices (HP2Vs) induced in pristine hBN thin flakes (Fig. 1a), associated with highly reconfigurable topological charges emerging at the HP2V center. Our system is composed of a gold (Au) disk with Archimedean spiral shape used to launch HP2 in hBN. We first study spin–orbit interactions in this system, revealing unprecedented control of HP2, the precise generation of nanoscale vortices with associated topological charges locked to the excitation spin. Based on scattering-type scanning near-field microscopy (s-SNOM) [13, 16, 42, 43], HP2Vs are observed in real space, and spin–orbit control and tunability are accordingly demonstrated. Our results reveal new degrees of freedom and enhanced robustness in HP2 control based on optical spin–orbit interactions and topological charges, ideally suited for super-resolution sensing and imaging, enhanced light-matter interactions, communications and multiplexing, and particle manipulation [44] at mid-IR frequencies.

Fig. 1
figure 1

Polariton vortex, dispersion, simulation and experimental results for an hBN flake excited through an Au spiral disk with m = 4. a Schematic of the geometry and experimental setup. b AFM image of Au disk shaped as a left-handed Archimedean spiral with four arms, rmax and rmin are the maximum and minimum radii, g is their difference, ϕ is the azimuthal angle and rAS(ϕ) is the distance from the center (O). c Dispersion of kt for the sample in panel b. Purple spheres and pink curve are experimental and calculated results, respectively. d–i Near-field amplitude images at ~ 1546 cm−1 (d and e), simulated Ez (f and g), and phase (h and i) distributions at ~ 1546 cm−1. Top and bottom images are obtained under RCP and LCP light illumination, respectively. The black dotted circle in the phase images indicates the spiral phase. The scale bar is 2 µm in all panels

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As schematically illustrated in Fig. 1a, our sample is composed of a thin hBN flake transferred onto a shaped Au disk. The edge of the nanodisk acts as an HP2 launcher, tailored to convert incident free-space light into strongly confined near fields at the disk boundary, resulting in efficient launching of HP2s [14, 15]. hBN flakes are known to support discrete waveguide modes based on bulk HP2s [13], denoted by the mode index s associated with their in-plane momentum of \({k}_{t,s}\), t refers to the axis parallel to the hBN flake. Unlike free-space beams or SPPs, HP2 propagation is characterized by a large wavevector k confined to the angle \(\theta =\mathrm{arctan}\left(\sqrt{-\frac{{\varepsilon }_{t}}{{\varepsilon }_{z}}}\right)\) with respect to the optical z axis, i.e., the axis perpendicular to the hBN flake, as illustrated in Additional file 1: Fig. S1. εz and εt are the hBN permittivity components along the z axis and in the orthogonal plane, respectively. Polariton fringes are expected to form with periodicity \({\delta }_{r}=2d\mathrm{tan}\theta\) on the top hBN surface, where d is the flake thickness.

In order to generate a polariton vortex in hBN, we control the optical path of the launched hyperbolic polaritons by tailoring the disk shape to follow an Archimedean spiral, \({r}_{AS}\left(\phi \right)={r}_{min}+\frac{\mathrm{mod}\left(m\phi , 2\pi \right)}{2\pi }*g\), where \(g\) indicates the rate of increase of the spiral radius with the azimuthal angle \(\phi\), and the integer parameter m denotes the number of spiral arms. As an example, a Au disk with outer shape mapping a four-arm spiral was fabricated, and its atomic force microscope (AFM) image is shown in Fig. 1b, where rmax and rmin are the maximum and minimum radius and \(g\equiv {r}_{max}-{r}_{min}\). Based on this design, the polaritons induced on the top surface are expected to support a phase distribution as a function of position (r,\(\phi\)) in the z = z0 plane following

$$\begin{array}{c}\varphi \left(r,\phi ,{z}_{0}\right)={k}_{t,s}\left({r}_{AS}\left(\phi \right)-{r}_{min}+\frac{{\delta }_{r}}{2}\right)+s{^{\prime}}2\pi \end{array}$$
(1)

where \(s{^{\prime}}\equiv s-1\) is the modified HP2 order on the flake with a Au substrate (see Sect. 4). As experimentally verified in Fig. 1, the induced HP2s indeed follow spiral orbitals in space to form a vortex with topological features. In particular, if the Archimedean spiral shape satisfies the condition

$${k}_{t,s}\left(g+\frac{{\delta }_{r}}{2}\right)=l2\pi ,$$
(2)

the phase accumulation from \(\phi =0\) to \(\phi =2\pi\) is \(m\left(l+s\mathrm{^{\prime}}\right)*2\pi\) for an Archimedean spiral with m arms. The integer l is the orbital topological charge supported by the induced HP2 spatial distribution.

Under a circularly polarized plane wave excitation, due to spin–orbit interactions in the near-field, the total phase accumulation is \(\left(l*m+\sigma \right)2\pi\), associated with a topological charge of total angular momentum

$$\begin{array}{c}L=m\left(l+s\mathrm{^{\prime}}\right)+\sigma ,\end{array}$$
(3)

where σ =  + 1 or − 1 or left-handed (LCP) or right-handed circularly polarized (RCP) incident light, as we assumed a left-handed spiral. Equation 3 outlines the several degrees of freedom available to control our HP2Vs, as a result of the complex interplay between incident spin and geometry-induced control of the polariton orbitals. Nontrivial control parameters consist of the spiral geometry (e.g., g), the working frequency (i.e., kt,s and l when g is fixed), the number of spiral arms m, the excitated polaritonic mode \(s\mathrm{^{\prime}}\), and the excitation spin \(\sigma\). In our demonstration, we focus on the dominant mode, s′ = 0.

HP2Vs carrying a discrete topological charge L are expected to support a spiral phase accumulation of L*2π and a phase singularity at their center [45], controlled by the spin of impinging light [29, 46,47,48]. As described by the previous analysis, this spin-dependent phase originates from spin-to-orbital angular momentum conversion enabled by the coupling between the spin angular momentum (SAM) of the excitation and the OAM [37]. To observe HP2Vs and the associated spin–orbit interactions in real space, we used s-SNOM measurements to map HP2s on the top surface of a hBN flake in Fig. 1a. The setup of our experiment is schematically shown in Fig. 2: we fabricated a Au four-arm Archimedean spiral disk over a SiO2 top layer on a silicon substrate, shown in Fig. 1b, with g = 650 nm, rmin = 2.275 μm and height h = 65 nm (details in Methods). Consistent with the previous analysis, the parameter h and rmin do not affect the total topological charge induced in the HP2V. An exfoliated hBN flake with a thickness of ~ 295 nm was transferred on top of the Au disk. By acquiring multiple near-field images with excitation frequency spanning from 1510 to 1570 cm−1 (see examples in Additional file 1: Fig. S2), we retrieved (Fig. 1c) the in-plane momentum (kt) of the fundamental HP2 mode at different frequencies, which agrees well with the analytical dispersion calculated based on the parameters in ref. [13].

Fig. 2
figure 2

Schematic of the s-SNOM (from neaspec GmbH) setup. QWP, BS, QCL, and RM represent the quarter-wave plate, beam splitter, quantum cascade laser, and reference mirror, respectively

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Figure 1d, e show the measured near-field distribution of polariton fringes at ~ 1546 cm−1, yielding a clear spiral shape profile emerging from the interference of HP2 fields launched by the disk boundary with the incident field at the tip. At this frequency, \({\lambda }_{t}\approx 761 \mathrm{nm}{=\lambda }_{0}/8.5\), where λt and λ0 are the in-plane and free-space wavelength, respectively, confirming a deeply subwavelength confinement of the induced HP2V. The simulated electric field (|Ez|) profile shown in Fig. 1f, g matches well with experimental results (see more details in Methods). Consistent with Eq. 3, when l = 1 a left-handed spiral with m = 4 yields polariton vortices with topological charge L = 3 and L = 5 for LCP and RCP excitation, respectively. By comparing Fig. 1d, e (or Fig. 1f, g), we notice a distinct difference in induced field profile at the vortex center, associated with the different topological charge. Both experimental and simulation results reveal that HP2V with L = 3 has a smaller ring-shaped profile at its center than L = 5, when l is fixed. This is expected, since its size increases with the spiral phase, consistent with other optical and plasmonic vortex studies [29, 45, 49]. Our setup imprints this dynamic into nanoscale phonon polaritons. It should be mentioned that the regions of the dotted circles in Fig. 1f–i and below figures are determined by the spiral phase at the center in near-field phase images and their positions are chosen to have a minimum deviation from their center to that of the Au patterns. The calculated phase distributions at ~ 1546 cm−1 under RCP and LCP excitation, shown in Fig. 1h (L = 3) and Fig. 1i (L = 5), reveal clockwise 3*2π and 5*2π spiral phase variations at the origin, further confirming the nature of spin–orbit interactions for σ =  + 1 and -1. The discrete topological charge associated with these polariton vortices introduces inherent robustness to geometric perturbations associated with topological features [50], and new degrees of freedom for polaritonic science and technology.

Next, we experimentally demonstrate precise reconfigurability of HP2Vs, besides the incident spin demonstrated in Fig. 1. According to Eq. 3, both orbit topological charge (l) and the Archimedean spiral geometry (m) control the total induced angular momentum of the polariton vortex. Here, l is controlled by the frequency of impinging light when g and d are fixed, following the polariton dispersion. In Fig. 3, using geometric arguments we predict the phase accumulation φPP along the azimuthal angle ϕ for m = 1, 2, 4 when l = 1, 2, revealing a broad range of opportunities in terms of polariton field profiles and induced topological charges that can be generated using this platform. In these plots, we do not explicitly take into account the additional degree of freedom stemming from the excitation spin, which enables an additional knob for manipulation. Because a large L leads to more dramatic amplitude and phase variations, and higher-order HP2Vs correspond to sharper field profiles, which are more difficult to image [30], in the following we demonstrate this broad reconfigurability for m = 1, 2 and l = 1, 2.

Fig. 3
figure 3

Polariton vortex phase accumulation vs azimuthal angle for m = 1, 2 and 4 when l = 1 and 2, and Au disk images and kt dispersion for polaritonic vortices with m = 1, 2. a, φPP and ϕ denote the accumulated phase and azimuthal angle, respectively. b,c AFM images of Au disk with one (b) and two (c) left-handed Archimedean spiral arms. d Dispersion of kt for samples in (b,c). Orange dots and navy curve are experimental and calculated results, respectively

Full size image

We first explore the near-field profile of the induced HP2Vs and their dependence on the number of Archimedean spiral arms m. Figure 3b, c show AFM images of Au disks with one- and two-arm Archimedean spirals, i.e., m = 1, 2, respectively. The parameters g, rmin, and h are the same as in the previous examples, and a thinner exfoliated hBN flake with d ~ 170 nm was transferred on top of the Au disks. Our experimentally measured polariton dispersion (see examples of amplitude images in Additional file 1: Fig. S3) agrees well with the analytical one also for this flake (Fig. 3d), showing larger transverse wavevectors, i.e., more confined polariton fields, compared to the results in Fig. 1c, due to the thinner nature of this flake. As seen in Eq. 2, larger wavevectors enable access to larger l for fixed g, hence supporting vortices with l = 1 and l = 2 around 1542 cm−1 and 1561 cm−1. Based on Eq. 3, for l = 1 we expect L = 0 (1) when m = 1 (2) under RCP light excitation, and L = 2 (3) when m = 1 (2) for LCP light.

The corresponding near-field images, showcasing the wide reconfigurability of polariton vortices enabled by these structures, are shown in Fig. 4. For L = 0, the phase distribution around the HP2V center is uniform, supporting a subwavelength polariton focal spot, rather than a ring profile as expected for nonzero topological charges (Fig. 4a). Figure 4e shows the corresponding numerical simulations, showing a good agreement. By varying the number of spiral arms or the excitation SAM, we can reconfigure the topological charge L. For example, increasing m from 1 to 2 makes L increase from 0 to 1, resulting in a different size of the central spot, which becomes ring-shaped and grows larger, as seen comparing Fig. 4a, g (Fig. 4c, i for simulations). Further growth of the central ring profile is obtained under LCP illumination (Fig. 4b and h), as this SAM yields larger values of L, again confirmed by our numerical simulations (Fig. 4d, j). It should be noticed that as revealed by the associated spiral phase in Fig. 4f the upper bright lobe close to the central amplitude profile in Fig. 4d is not a part of the central pattern while at an outer layer of the spiral pattern, so L is 2 not 3 in Fig. 4d. As L increases, we also verify that the polariton vortex phase accumulation grows in discrete 2π steps, as illustrated in Fig. 4e, f, k, l. We find 0 to 3*2π spiral phase accumulation around the center, confirming a discrete growth of topological charge L from 0 to 3. These images also show a 2*2π phase difference between LCP and RCP excitation, demonstrating strong spin–orbit polariton interactions.

Fig. 4
figure 4

Experimental and simulation results for polariton vortices when l = 1. a, b, g, h Near-field amplitude images at ~ 1550 cm−1 for right-handed (a and g) and left-handed (b and h) circularly polarized light. c, d, i, j Numerical Ez and e, f, k, l phase distributions at ~ 1542 cm−1 for right-handed (c, i, e, and k) and left-handed (d, j, f, and l) circularly polarized light. The zoom-in images of the central profile are in the lower column. The cyan/red dotted circles with numbers in the zoom-in amplitude/simulated Ez images label the central interference profiles. The black dotted circles in the phase images with numbers indicate the spiral phases. The scale bars are 2 µm in the full images and 0.67 µm in the zoom-in ones

Full size image

Extreme HP2V reconfigurability can also be achieved by controlling the orbital topological charge l, which can be tuned through the excitation frequency. Figure 5 shows results with a similar trend as in Fig. 4 retrieved from the same samples at ~ 1561 cm−1, for which the increased kt of the supported polaritons yields an orbital topological charge l = 2. As a result, the total angular momentum of HP2V correspondingly increases compared to the l = 1 scenario. Since HP2s support a broad range of frequency-dependent kt thanks to their hyperbolic dispersion [11, 13], a wide range of l can be attained, enabling broad reconfigurability of HP2Vs without varying the sample geometry. The demonstrated nano-vortices have deeply subwavelength spatial profile, again supported by the extreme HP2 field confinement. A comparison between experimental and simulated images in Figs. 4 and 5 shows that even denser fringes are obtained when l = 2, consistent with the larger kt and shorter polariton wavelength (Eq. 1) combined with increased topological charge. At ~ 1550 cm−1 (Fig. 4a, b, g, h), \({\lambda }_{t}\approx {\lambda }_{0}/14.8\) (l = l), while at ~ 1566 cm−1 (Fig. 5a, b, g, h) \({\lambda }_{t}\approx {\lambda }_{0}/19.5\) (l = 2). The near-field profiles at the vortex center are supported through the superposition of HP2s excited from the Au disk boundary, so we can expect that the size of the HP2V central profile narrows as l increases. To further explore how the phase profile evolves with l, we compare Fig. 4h, j, and l (l = 1) with Fig. 5g, i, and k (l = 2). A sharper phase variation and a smaller central profile can indeed be observed at the higher frequency, despite the fact that in both scenarios the induced topological charge is the same (L = 3). For the L = 5 scenario when l = 2 in Fig. 5l the spiral phase is less well defined compared to the one in Fig. 1i, associated with the fact that the spiral phase is less stable at larger L when l = 2 than l = 1. Overall, Fig. 5 features HP2s with very large transverse momentum, realizing polariton vortices with highly confined features, offering unprecedented opportunities for super-resolution mid-IR imaging, high-precision particle manipulation, sensing, and high-density information storage and transmission. When l is larger than 2 the associated frequencies become too close to the 1610 cm−1, i.e., the high-frequency edge of the upper Reststrahlen band of [13]. Therefore, it is not possible to obtain a clean central profile in this scenario, because of increased loss and reduced group velocity of the associated polaritons (more discussions can be found in the Additional file 1).

Fig. 5
figure 5

Experimental and simulation results for polariton vortices when l = 2. a, b, g, h Near-field amplitude maps at ~ 1566 cm−1 for right-handed (a and g) and left-handed (b and h) circularly polarized light. c, d, i, j Numerical Ez, and e, f, k, l phase distributions at ~ 1561 cm−1 for right-handed (c, i, e, and k) and left-handed (d, j, f, and l) circularly polarized light. The zoom-in images of the central profile are in the lower column. The cyan/red dotted circles with numbers in the zoom-in amplitude/simulated Ez images label the central interference profiles. The black dotted circles in the phase images with numbers indicate the spiral phases. The scale bars are 2 µm in the full images and 0.67 µm in the zoom-in ones

Full size image

During the review of this study, Xiong et al. reported topological spin textures of phonon polaritons by illuminating a ring-shaped antenna located on top of an hBN flake with circularly polarized light [51]. In their study, meron-like textures with a half-integer topological charge, determined by the handedness of the incident circularly polarized light, were observed, linking transverse SAM to skyrmion/meron-like textures in polaritons [51,52,53]. Here, we explore the spin–orbit interaction between OAM and out-of-plane SAM and the control of total topological charges via multiple degrees of freedom.

3 Conclusions

Overall, in this work we have conclusively demonstrated spin–orbit locked HP2Vs emerging in natural polar vdW crystal hBN flakes coupled to spiral structures with tailored in-plane chirality. The resulting devices generate broadly reconfigurable topological charges, showcasing exotic polaritonic features, including spin–orbit interactions and nanofocusing. The demonstrated polariton vortices are highly tunable using various degrees of freedom, including the excitation spin, the geometry of the polariton launcher, and the hyperbolic features of the underlying material controlled by the wavelength of excitation. Our demonstration of HP2V opens unique opportunities to robustly process multiplexed information at mid-IR wavelengths and has great potential for super-resolution imaging systems with multiplexing capabilities, ultracompact mid-IR sensors, and miniaturized polaritonic devices with robust features associated with their topological nature, as well as enhanced nonlinearities and sensing. Overall, our findings broadly enrich the nanophotonic and polaritonic platforms enabled by vdW nanomaterials with phase, spin, and orbital angular momentum engineering capabilities, of great interest for a variety of classical and quantum applications.

4 Methods

4.1 Sample preparations

Gold (Au) disks with Archimedean spiral shapes were prepared through standard electron beam lithography (EBL). EBL resist (PMMA) was first spin-coated on a silicon dioxide (SiO2) layer on a silicon substrate, and the desired patterns were created by exciting the resist with the electron beam. After resist developing in an MIBK-IPA solution, a 65-nm-thick Au layer was deposited on top of the sample via electron beam evaporation, and the resist was lifted off. Atomic force microscope (AFM) images of the realised Au disks are shown in Figs. 1b and 3b, c. Hexagonal boron nitride (hBN) flakes were obtained through mechanical exfoliation of high-quality hBN crystals. hBN flakes were transferred on top of the Au disks via a dry transfer method, as shown in Additional file 1: Fig. S4.

4.2 Near-field measurements

A commercial scattering-type scanning near-field optical microscope (s-SNOM) (neaspec GmbH) was used to record the near-field images of the hyperbolic polaritonic vortex (HP2V) in hBN. A quantum cascade laser (QCL) with tunable wavelength (MIRcat-QT by DRS Daylight Solutions) was employed as the mid-infrared light source. The s-SNOM measurement was based on a tapping-mode AFM, and a commercial AFM tip with PtIr5 coating (NanoWorld) was utilized to probe the local near field on the top of the hBN flakes. The scattering signals from the tip were recorded by a pseudo-heterodyne interferometric detection module with AFM tapping amplitude of ~ 70 nm at a fixed wavelength. The obtained signals were demodulated at the 2nd harmonic of the oscillation frequency, which is about 285 kHz. A schematic presentation of the setup is shown in Fig. 2.

As shown in Fig. 2, a circularly polarized mid-IR beam was produced by passing a linearly polarized beam through a mid-IR quarter-wave plate, and then focused on the sample with a parabolic mirror. In order to verify whether the beam focused by a parabolic mirror still preserves circular polarization, we compared the output intensities of two setups: with and without a parabolic mirror, as schematically illustrated in Additional file 1: Fig. S5. The results obtained at 1510 cm−1 are listed in Additional file 1: Tables S1 and 2, which prove the preservation of circular polarization after the beam is focused by the parabolic mirror. An Au disk without spiral arms covered by an hBN flake was then measured to confirm that HP2 with σ =  + 1 and -1 can be excited under LCP and RCP incident light, respectively. The results are shown in Additional file 1: Fig. S7 and are in good agreement with the simulations. In addition, Additional file 1: Fig. S8 unveils that there are important differences between the amplitude images retrieved under linearly and circularly polarized light. Please see more discussions in Additional file 1.

4.3 Theoretical analysis and numerical simulations

For a hBN thin layer on top of a perfect electric conductor (PEC) substrate, the phonon polariton (PhP) mode in hBN can be obtained by finding the poles of the dyadic Green’s function, which is equivalent to the transverse resonance condition:

$$2{k}_{z,s}d+\angle {R}_{1}+\angle {R}_{2}=s2\pi ,\ s= 0,1,2,\ldots$$
(4)

where d is the thickness of hBN, \({R}_{1}=-\frac{{\varepsilon }_{1}{k}_{z}-{\varepsilon }_{t}{k}_{z1}}{{\varepsilon }_{1}{k}_{z}+{\varepsilon }_{t}{k}_{z1}}\) and \({R}_{2}=\frac{{\varepsilon }_{2}{k}_{z}-{\varepsilon }_{t}{k}_{z2}}{{\varepsilon }_{2}{k}_{z}+{\varepsilon }_{t}{k}_{z2}}\) are the reflection coefficient at the top surface and bottom surface; The top and bottom material are assumed to have permittivity \({\varepsilon }_{1}\) and \({\varepsilon }_{2}\), respectively. Here, \({k}_{z1}=\sqrt{{\varepsilon }_{1}{k}_{0}^{2}-{k}_{t}^{2}}\); \({k}_{z2}=\sqrt{{\varepsilon }_{2}{k}_{0}^{2}-{k}_{t}^{2}}\); and \({k}_{z,s}=\sqrt{{\varepsilon }_{t}{k}_{0}^{2}-\frac{{\varepsilon }_{t}}{{\varepsilon }_{z}}{k}_{t}^{2}}\), where \({k}_{t}\) is the in-plane wavevector and the index s denotes the s-th waveguide mode. For large wavevector, we have:

$${k}_{z,s}\approx {k}_{t,s}\sqrt{-\frac{{\varepsilon }_{t}}{{\varepsilon }_{z}}}={k}_{t,s}\mathrm{tan}\theta$$
(5)

i.e., \({k}_{t,s}=\frac{{k}_{z,s}}{\mathrm{tan}\theta }\), where θ is the angle with respect to the optical axis (z axis) for a Type II hyperbolic band. A fringe at the top surface can be formed with evolving periodicity \({\delta }_{r}\equiv 2d\mathrm{tan}\theta\). Alternatively, Eq. (4) can be written as

$${k}_{t,s}=-\frac{\Psi }{d}\left[\mathrm{arctan}\left(\frac{{\varepsilon }_{1}}{{\varepsilon }_{t}}\Psi \right)+\mathrm{arctan}\left(\frac{{\varepsilon }_{2}}{{\varepsilon }_{t}}\Psi \right)+s\pi \right]$$

where \(\Psi \equiv \frac{\sqrt{{\varepsilon }_{z}}}{i\sqrt{{\varepsilon }_{t}}}\). If the substrate is gold at mid-infrared frequency, \(\mathrm{Re}\left[{\varepsilon }_{2}\right]\)<0 and \(\left|{\varepsilon }_{2}\right|\gg\) 0, we thus have \(\mathrm{arctan}\left(\frac{{\varepsilon }_{2}}{{\varepsilon }_{t}}\Psi \right)\to -\frac{\pi }{2}\) and \(\left|{R}_{2}\right|\to 1\) while \(\angle {R}_{2}\to -2\pi\). Therefore, the lowest-order mode is s = 1. This differs from the case of dielectric substrates with positive permittivity. Nevertheless, one can rewrite this mode as

$${k}_{t,s}=-\frac{\Psi }{d}\left[\mathrm{arctan}\left(\frac{{\varepsilon }_{1}}{{\varepsilon }_{t}}\Psi \right)+\left(s-1\right)\pi \right]=-\frac{\Psi }{d}\left[\mathrm{arctan}\left(\frac{{\varepsilon }_{1}}{{\varepsilon }_{t}}\Psi \right)+{s}{^{\prime}}\pi \right]$$

where we defined \({s}{^{\prime}}\equiv s-1\) for brevity. In order to form a vortex, we design the Au disk to have an Archimedean spiral shape, which can be mathematically written as \({r}_{AS}\left(\phi \right)={r}_{0}+m\frac{\mathrm{mod}\left(\phi , 2\pi \right)}{2\pi }g\), where \({r}_{0}\) is the minimum value of the radius and \(g\) is a geometric parameter indicating the rate of increase of the radius with the azimuthal angle ϕ, defined as \(g={r}_{AS}\left(\phi =2\pi \right)-{r}_{0}\). The integer parameter m denotes the number of consecutive Archimedean arms in the pattern. We can calculate the phase of the polariton distribution at the top surface, given by the evolution of the polaritons launched from the edge:

$$\begin{array}{c}\varphi \left(r,\phi \right)={k}_{t,s}\left({r}_{AS}-r\right)+{k}_{z,s}d+mod\left({r}_{AS}-r-d\mathrm{cot}\theta ,2d\mathrm{cot}\theta \right)\cdot \left(\angle {R}_{1}+\angle {R}_{2}\right),\end{array}$$
(6)

where the first and second term refer to the propagation phase of polaritons. The third term refers to the phase accumulation due to the reflection from the top and bottom surface, which should also consider a reflection phase shift at the substrate, since what is measured is the polariton distribution at the top. Considering the quantized model of the polariton waveguide in Eq. 4 and the dispersion relation in Eq. 5, we could further simplify this expression as

$$\begin{array}{c}\varphi \left(r,\phi \right)={k}_{t,s}\left({r}_{AS}-r+\frac{{\delta }_{r}}{2}\right)+{s}{\mathrm{^{\prime}}}2\pi . \end{array}$$
(7)

If we design the Archimedean spiral such that \({k}_{t,s}\left({r}_{AS}-r+\frac{{\delta }_{r}}{2}\right)=2\pi l\), the phase accumulation of one Archimedean spiral is \(\left(l+s{^{\prime}}\right)2\pi\). Therefore, the propagation phase accumulation after a rotation from \(\phi =0\) to \(\phi =2\pi\) is

$${\varphi }_{PP}=\left(l+s{^{\prime}}\right)\cdot m2\pi$$
(8)

Here, we focus on the s′ = 0 mode, which is also more easily excited because it has smaller momentum mismatch with free-space plane waves. Under a circularly polarized plane wave excitation, due the spin–orbit interactions in the near-field, we have the total phase accumulation

$${\varphi }_{PP}=\left(l+s{^{\prime}}\right)\cdot m2\pi +\sigma 2\pi$$
(9)

which yields the total topological charge

$$L=\left(l+s{^{\prime}}\right)\cdot m+\sigma$$
(10)

where s′ = 0 is the dominant mode. For left-handed structures, \(\sigma =\) +1 or −1 for left-handed circularly polarized (LCP) or right-handed circularly polarized (RCP) light, respectively.

We performed full-wave numerical simulations using the finite-difference time-domain commercially-available software Lumerical FDTD (https://www.lumerical.com/products/fdtd/). Two orthogonal linearly polarized waves with phase difference ± π/2 were used to generate the circularly polarized excitation. An incident angle of 10°, which is chosen for achieving the best agreement between experimental results and simulations, is applied because an off-axis parabolic mirror is used to focus the beam on the samples in the experiments. A monitor was placed at 20 nm above the top surface to calculate the near-field images.