Higher-Order Topological Acoustic Vortices

• Higher-Order Topological Charge Acoustic Vortices are acoustic wavefields with quantized phase windings that impart discrete orbital angular momentum for advanced signal encoding.
• Device engineering leverages shaped resonators and metamaterial apertures to dynamically tune vortex charge and enable multimodal operation across GHz frequencies.
• Interferometric and hydrophone measurements confirm modal orthogonality and skyrmionic spin textures, underpinning applications in particle manipulation and hybrid photonic systems.

Higher-order topological charge acoustic vortices are wavefields in acoustics characterized by quantized phase windings around a central singularity, encapsulated by an integer topological charge $\ell$, and supporting robust orbital angular momentum (OAM) transport. Recent advances leverage engineered resonator geometries, metamaterial apertures, and acousto-optic coupling to dynamically generate and manipulate such vortices across a wide frequency range (from kHz to several GHz), enabling diverse applications in signal multiplexing, particle manipulation, and angular-momentum photonics. Core phenomena include the evolution of quantized/nonquantized phase singularities, the emergence of skyrmionic spin textures, and modal orthogonality critical for parallel information encoding.

1. Theoretical Framework for Acoustic Vortices

Acoustic vortex modes are defined by solutions to the wave equation in cylindrical coordinates. For an isotropic medium, the displacement field $\vec{u}(\rho,\phi,z,t)$ for a mode of order $m$ takes the form $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$ (Pitanti et al., 2024). The phase winding number $m$ determines the topological charge: $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$, so the phase increases by $2\pi m$ along any closed contour encircling the vortex core. Each phonon in such a mode carries $m\hbar$ mechanical OAM about the $z$-axis.

The velocity and pressure fields associated with higher-order vortices are commonly decomposed via Bessel functions: $$u_\rho(\rho,\phi,t) = U_0\,J_m(k_\rho \rho)\,e^{i(m\phi-\omega t)}$$, with $\vec{u}(\rho,\phi,z,t)$0 the $\vec{u}(\rho,\phi,z,t)$1 order Bessel function and $\vec{u}(\rho,\phi,z,t)$2 the in-plane wavevector. This formulation underpins both bulk acoustic-wave resonator (BAWR) (Pitanti et al., 2024) and metamaterial aperture designs (Naify et al., 2016).

2. Device Engineering and Charge Tunability

a) Shaped Resonators

Shape-engineering is essential for higher-order vortex generation. The Archimedean spiral launcher in a single-contact BAWR utilizes a top contact defined by $\vec{u}(\rho,\phi,z,t)$3, with radial and vertical confinement dictated by substrate and piezoelectric layer geometry. Tuning the charge $\vec{u}(\rho,\phi,z,t)$4 exploits the drive frequency $\vec{u}(\rho,\phi,z,t)$5 to alter the in-plane wavelength $\vec{u}(\rho,\phi,z,t)$6, leading to $\vec{u}(\rho,\phi,z,t)$7 with $\vec{u}(\rho,\phi,z,t)$8 fixed by spiral geometry. Experimentally, $\vec{u}(\rho,\phi,z,t)$9 from $m$0 to $m$1 was directly measured up to $m$2 GHz, with simulations indicating robust vortex formation up to $m$3 at $m$4 GHz (Pitanti et al., 2024).

Multi-spoke spirals generalize the charge scaling, following $m$5, so the resonance becomes $m$6, permitting $m$7 at fixed $m$8. This enables integer scaling and parallel generation of higher $m$9 modes.

b) Metamaterial Apertures

The metamaterial annular leaky-wave antenna (Naify et al., 2016) achieves topological charge tunability via frequency-controlled phase winding. The effective refractive index $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$0 defines the phase accumulation $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$1 with $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$2. Integer $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$3 yields pure single-singularity vortices; fractional $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$4 corresponds to multi-singularity dislocation patterns. Efficient mode separation, evidenced by low cross-talk ($$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$5 for $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$6), ensures suitability for OAM-multiplexed applications.

3. Measurement, Characterization, and Core Spin Texture

a) Interferometric and Hydrophone Probing

Complex displacement and phase maps of acoustic vortex fields are captured via optical interferometry (Michelson, $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$7 nm, $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$81 $$\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$$9m spot size) in BAWRs (Pitanti et al., 2024). In metamaterial apertures, phase maps at distances above the surface reveal $m$0 to $m$1 full $m$2 phase wraps for corresponding $m$3, reproducing simulation results (Naify et al., 2016). Pressure amplitude profiles exhibit central nulls with radius increasing with $m$4.

In 3D hydrophone array studies, the velocity field $m$5 is reconstructed, permitting full analysis of local spin. The canonical spin density $m$6 captures the local angular momentum content (Annenkova et al., 2 Dec 2025).

b) Skyrmionic Spin Topology

Vortex cores universally support spin merons (half-skyrmions): the spin unit vector in acoustics $m$7 with $m$8 yields a meron with half-integer Skyrme number $m$9 (Annenkova et al., 2 Dec 2025). For first-order ($$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$0), this topology is nondiffractive and robust against propagation, modal basis, and phase sampling. For higher orders ($$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$1), universality breaks down: phase singularities split (Nye–Berry instability), generating multiple half-skyrmions in the core—experimentally evidenced as triplet textures for $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$2.

4. Acousto-Optic Modulation via Vortex Beams

The acousto-optic interaction is mediated by periodic modulation of the dielectric/air interface through the surface displacement $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$3 (Pitanti et al., 2024). Incident probe light is phase-modulated by the dynamic surface corrugation, such that the angular phase term $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$4 in the acoustic field imprints an OAM structure onto the optical reflection. The governing relation for the optical field modulation at each point is $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$5, with sidebands acquiring $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$6 angular momentum per photon. The sign of OAM is set by propagation direction.

This principle enables electrically driven, dynamically tunable OAM modulation at GHz rates, with on-chip integration for hybrid photonic-acoustic systems.

5. Modal Orthogonality, Charge Scalability, and Applications

Orthogonality between integer-$$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$7 vortex modes is critical for multiplexed information transmission and manipulation. Covariance analysis yields $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$8 for $$\ell = m = (1/2\pi)\oint_C \nabla\phi \cdot d\mathbf{r}$$9 (Naify et al., 2016), underscoring the ability to multiplex OAM channels with minimal crosstalk.

The operating frequency spans $2\pi m$0–$2\pi m$1 GHz (BAWR), with practical $2\pi m$2 under current phase-mapping resolution, scaling to $2\pi m$3 in simulation or with atomic-force imaging. In metamaterial designs, achievable $2\pi m$4 is limited by guided-mode cutoff and aperture size; compactness ($2\pi m$5 at high $2\pi m$6) supports dense integration and high-order vortex emission.

Applications include:

• On-chip GHz-rate OAM modulation for high-capacity optical communications (Pitanti et al., 2024)
• Acoustic tweezing and particle manipulation, exploiting vortex orbital motion and torque (Naify et al., 2016)
• Hybrid phonon-photon devices with angular-momentum-based control of exciton-polaritons, magnonics, and quantum acoustodynamics
• Encoding, storage, and manipulation of skyrmionic spin textures for information processing (Annenkova et al., 2 Dec 2025)

6. Limitations, Universality Breakdown, and Future Directions

For higher-order charges ($2\pi m$7), universality of core spin topology is contingent on mode structure and phase stability. In acoustics, core topology is robust, but in optics, Bessel-mode analysis reveals breakdown due to spin-orbit coupling, generating integer Skyrme charges and extra longitudinal-transverse mixing (Annenkova et al., 2 Dec 2025). Experimental data show core splitting into multiple charges beyond $2\pi m$8, marking the onset of Nye–Berry singularity separation.

A plausible implication is that device architectures exploiting higher-order acoustic vortices must carefully account for modal purity, stability of the phase singularity, and detection resolution. Limitations in modal cutoff, substrate resonance $2\pi m$9, and mapping resolution define the practical $m\hbar$0 upper bound in each architecture.

Emergent directions focus on integration with metamaterials, ultrafast photonic-audio modulation, and topologically encoded acoustic manipulation at subwavelength scales, with ongoing research into robust control of skyrmionic core structures for high-density information technologies.

7. Overview of Experimental and Simulation Methods

Architecture	Modal Basis	Tunable $m\hbar$1 Range
BAWR (spiral launcher)	Cylindrical Bessel	1–4 (exp.), up to 13 (sim.)
Metamaterial aperture	Annular leaky-wave	0–3 (exp.), higher possible
Hydrophone array	LG/Bessel	$m\hbar$2 to $m\hbar$3 (sectorized)

Measurement protocols employ frequency sweeps to control $m\hbar$4 and $m\hbar$5, simulation via 3D FEM modeling (COMSOL) to predict amplitude/phase map evolution, and interferometric and hydrophone-based rastering for displacement and velocity field characterization. All systems demonstrate dynamic, electrically and geometrically controlled generation of higher-order topological acoustic vortices, with simulation and experiment in quantitative agreement for field topology, amplitude nodal structure, and phase singularity evolution (Pitanti et al., 2024, Naify et al., 2016, Annenkova et al., 2 Dec 2025).