---
title: Higher-Order Topological Acoustic Vortices
url: https://www.emergentmind.com/topics/higher-order-topological-charge-acoustic-vortices
type: topic
---

# Higher-Order Topological Acoustic Vortices

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 $\vec{u}(\rho,\phi,z,t) = [u_\rho(\rho)\mathbf{e}_\rho + u_\phi(\rho)\mathbf{e}_\phi + u_z(\rho)\mathbf{e}_z]\,e^{i(m\phi + k_z z - \omega t)}$ [2410.17877]. 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 $J_m$ the $m^\text{th}$ order Bessel function and $k_\rho = 2\pi/\lambda$ the in-plane wavevector. This formulation underpins both bulk acoustic-wave resonator (BAWR) [2410.17877] and metamaterial aperture designs [1604.08447].

## 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 $AS(\theta) = R_0 + (g/2\pi)\cdot\theta$, with radial and vertical confinement dictated by substrate and piezoelectric layer geometry. Tuning the charge $\ell$ exploits the drive frequency $f$ to alter the in-plane wavelength $\lambda$, leading to $g/\lambda \simeq m$ with $g$ fixed by spiral geometry. Experimentally, $\ell$ from $1$ to $4$ was directly measured up to $1.3$ GHz, with simulations indicating robust vortex formation up to $\ell \approx 13$ at $5$ GHz [2410.17877].

Multi-spoke spirals generalize the charge scaling, following $Sp_M(\theta) = R_0 + (\mathrm{mod}(M\theta,2\pi)/2\pi)\cdot g$, so the resonance becomes $M(g/\lambda) = m$, permitting $\ell = M(g/\lambda)$ at fixed $f$. This enables integer scaling and parallel generation of higher $\ell$ modes.

### b) Metamaterial Apertures

The metamaterial annular leaky-wave antenna [1604.08447] achieves topological charge tunability via frequency-controlled phase winding. The effective refractive index $n(\omega) = \beta(\omega)/k(\omega)$ defines the phase accumulation $\Delta\Phi = \beta(\omega)\cdot 2\pi R_m$ with $\ell(\omega) = \beta(\omega)R_m$. Integer $\ell$ yields pure single-singularity vortices; fractional $\ell$ corresponds to multi-singularity dislocation patterns. Efficient mode separation, evidenced by low cross-talk ($|C_{mn}| \lesssim 0.05$ for $m \ne n$), 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, $\lambda=532$ nm, $\sim$1 $\mu$m spot size) in BAWRs [2410.17877]. In metamaterial apertures, phase maps at distances above the surface reveal $0$ to $\pm3$ full $2\pi$ phase wraps for corresponding $\ell$, reproducing simulation results [1604.08447]. Pressure amplitude profiles exhibit central nulls with radius increasing with $|\ell|$.

In 3D hydrophone array studies, the velocity field $\vec{v}(\mathbf{r}) = \nabla p(\mathbf{r})/(i \varrho \omega)$ is reconstructed, permitting full analysis of local spin. The canonical spin density $\mathbf{S}(\mathbf{r}) = \Im[\mathbf{v}^*(\mathbf{r}) \times \mathbf{v}(\mathbf{r})]/|\mathbf{v}(\mathbf{r})|^2$ captures the local angular momentum content [2512.02964].

### b) Skyrmionic Spin Topology

Vortex cores universally support spin merons (half-skyrmions): the spin unit vector in acoustics $\mathbf{\widetilde s}_{\rm sound}(\rho) = (0,-1,\ell\xi)/\sqrt{1+\ell^2\xi^2}$ with $\xi=(k\rho)^{-1}$ yields a meron with half-integer Skyrme number $N=\pm1/2$ [2512.02964]. For first-order ($|\ell|=1$), this topology is nondiffractive and robust against propagation, modal basis, and phase sampling. For higher orders ($|\ell|>1$), universality breaks down: phase singularities split (Nye–Berry instability), generating multiple half-skyrmions in the core—experimentally evidenced as triplet textures for $\ell=2$.

## 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 $u_z(\rho,\phi,t)$ [2410.17877]. Incident probe light is phase-modulated by the dynamic surface corrugation, such that the angular phase term $m\phi$ in the acoustic field imprints an OAM structure onto the optical reflection. The governing relation for the optical field modulation at each point is $\Delta\psi(\rho,\phi,t) = k_\mathrm{opt}\,u_z(\rho,\phi,t)$, with sidebands acquiring $\ell_\mathrm{opt} = \pm m$ 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-$\ell$ vortex modes is critical for multiplexed information transmission and manipulation. Covariance analysis yields $|C_{mn}| \lesssim 0.05$ for $m \ne n$ [1604.08447], underscoring the ability to multiplex OAM channels with minimal crosstalk.

The operating frequency spans $0.5$–$7$ GHz (BAWR), with practical $\ell_\mathrm{max} \approx 4$ under current phase-mapping resolution, scaling to $\ell \gtrsim 13$ in simulation or with atomic-force imaging. In metamaterial designs, achievable $\ell$ is limited by guided-mode cutoff and aperture size; compactness ($\sim \lambda$ at high $f$) supports dense integration and high-order vortex emission.

Applications include:
- On-chip GHz-rate OAM modulation for high-capacity optical communications [2410.17877]
- Acoustic tweezing and particle manipulation, exploiting vortex orbital motion and torque [1604.08447]
- 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 [2512.02964]

## 6. Limitations, Universality Breakdown, and Future Directions

For higher-order charges ($|\ell|>1$), 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 [2512.02964]. Experimental data show core splitting into multiple charges beyond $|\ell|=1$, 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 $Q$, and mapping resolution define the practical $|\ell|$ 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 $\ell$ 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         | $-3$ to $+3$ (sectorized)  |

Measurement protocols employ frequency sweeps to control $\lambda$ and $\ell$, 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 [2410.17877][1604.08447][2512.02964].

Source: https://www.emergentmind.com/topics/higher-order-topological-charge-acoustic-vortices