Toroidal Pulses: Theory & Applications

• Toroidal pulses are electromagnetic and hydrodynamic waveforms characterized by a doughnut-shaped field topology and intrinsic space–time nonseparability.
• They are modeled by exact solutions to Maxwell’s and Navier–Stokes equations, revealing phenomena like skyrmionic textures and vortex streets.
• Advanced generation techniques using metasurfaces and ultrafast pulse shaping enable robust light–matter interactions for imaging, communications, and quantum technologies.

Toroidal pulses are a class of electromagnetic or hydrodynamic waveforms characterized by toroidal topology, nontrivial spatiotemporal and polarization structure, and unique propagation invariance or dynamic features. Originating from both plasma physics and photonics, they encompass vector electromagnetic pulses with doughnut-shaped energy density, pulses with predominantly longitudinal fields, supertoroidal and hybrid vortex configurations, as well as hydrodynamic pulses in curved geometries. Their unifying feature is a field topology in which phase, polarization, or current forms a toroidal (doughnut-like) arrangement, leading to phenomena such as skyrmionic textures, space–time nonseparability, topological protection, and novel light–matter interaction regimes.

1. Mathematical Foundations and Field Topology

Toroidal pulses are exact solutions to their underlying wave equations—Maxwell’s equations for light, the Navier–Stokes equations for hydrodynamics, or the appropriate quantum equations in condensed matter. A prototypical example is the canonical flying doughnut pulse, whose solution in cylindrical coordinates $(\rho, \theta, z)$ is given by

$$E_\theta = -4i f_0 \sqrt{\mu_0/\epsilon_0} \frac{\rho (q_2-q_1-2i c t)}{\left[ \rho^2 + (q_1 + i\tau)(q_2 - i\sigma) \right]^3 },$$

with $\tau = z - ct$, $\sigma = z + ct$, and geometric parameters $q_1$, $q_2$ controlling the effective wavelength and focal region. The field exhibits an azimuthal electric component ($E_\theta$) with a meridional magnetic field, together forming a toroidal structure.

In more advanced variants—the supertoroidal light pulses (STLPs)—the solution is parametrized by an order parameter $\alpha \geq 1$:

$$f(r, t) = \frac{f_0}{(q_1 + i\tau) (s + q_2)^\alpha}, \quad s = \frac{r^2}{q_1 + i\tau} - i\sigma,$$

together with the Hertz potential construction $A = \mu_0 (\nabla \times \hat{z} f(r, t))$, yielding electromagnetic fields with multi-shell singularities, fractal-like structure, and hierarchical skyrmion organizations (Shen et al., 2021, Shen et al., 2022).

Scalar and vector hybrid toroidal vortices, mathematically mediated by designed transfer functions on metasurfaces, interlink the singularities of scalar (phase) and vector (field) landscapes (Wang et al., 20 Aug 2024). Recent theoretical frameworks also introduce chiral extensions (helical pulses) of toroidal pulses by incorporating angular momentum via a phase factor $e^{i \ell \phi}$ in the generating function, giving rise to double-helix field singularities (Shi et al., 20 Sep 2024).

In hydrodynamics, the time-periodic toroidal pulses in pipe flow arise as fixed points of the Fourier-transformed Navier–Stokes system, with the dynamics encoded in the truncated modal decomposition of the velocity and pressure fields (Lupi et al., 2023).

2. Space–Time and Space–Polarization Nonseparability

A defining feature of electromagnetic toroidal pulses is their space–time (or space–frequency) nonseparability. Unlike conventional pulses, where the field factors into spatial and temporal components, toroidal pulses couple these degrees of freedom intrinsically:

$E(x, t) \neq A(x) \cdot T(t).$

This space–time entanglement enables each spectral component to be localized at a distinct spatial position, producing frequency-dependent beam profiles and isodiffracting propagation; all monochromatic constituents share the same Rayleigh range and maintain fixed relative spatial positions during propagation (Zdagkas et al., 2019, Shen et al., 2020, Zdagkas et al., 2021, Vignjevic et al., 13 Sep 2025).

Space–polarization nonseparability also arises, as the polarization state varies across the spatial profile in a manner not reducible to an independent polarization vector and spatial amplitude. This is explored via state decomposition and higher-order Poincaré sphere formalism:

$$|\Psi\rangle = a |U_R, x\rangle + b e^{i\phi} |U_L, y\rangle,$$

with $|U_{R/L}\rangle$ denoting circular polarization and $|x/y\rangle$ spatial modes (Vignjevic et al., 13 Sep 2025).

3. Topological Features: Skyrmions, Vortex Streets, and Singularities

Toroidal pulses host rich topological structures. Skyrmionic textures—vector field configurations with nontrivial winding—are ubiquitous. These are parametrized by skyrmion number

$$s = \frac{1}{4\pi} \int \mathbf{n} \cdot (\partial_x \mathbf{n} \times \partial_y \mathbf{n}) dx\,dy,$$

where $\mathbf{n} = \mathbf{H} / |\mathbf{H}|$ for the magnetic field. Skyrmions can be of Néel, Bloch, or anti-skyrmion type, depending on the field mapping from physical to vector space (Shen et al., 2021, Wang et al., 2023).

ND-STPs and supertoroidal pulses reveal fractal-like patterns of phase singularities (zero scalar field) in nested shells, matryoshka-like singular structures, and periodic arrays of vortex dipoles forming optical "Kármán vortex streets"—three-dimensional analogues of fluid-dynamic vortex patterns (Shen et al., 2022). Chiral and helical toroidal pulses display persistent double-helix field singularities and dynamic vortex–antivortex annihilation events during propagation (Shi et al., 20 Sep 2024, Wang et al., 14 Jan 2025).

Hybrid pulses generated via metasurfaces combine scalar phase vortices (OAM $=\pm1$) with vector saddle points, giving rise to electromagnetic vortex streets and robust skyrmion patterns (Wang et al., 20 Aug 2024).

4. Generation, Manipulation, and Characterization Techniques

Generation methods include:

• Ultrafast pulse shaping on metasurfaces patterned with concentric rings, Y-shaped meta-atoms, or split-ring resonators for both optical and terahertz domains (Zdagkas et al., 2021, Niu et al., 29 Aug 2025).
• Singular metamaterial converters with spatially varied resonant dipoles that couple the spatial and spectral properties of a conventional pulse to synthesize flying doughnut states (Papasimakis et al., 2017).
• Compact antenna structures (coaxial horn, spiral emitter) combined with metasurface phase manipulation in microwaves (Wang et al., 2023, Wang et al., 20 Aug 2024, Wang et al., 14 Jan 2025).
• Hydrodynamic and computational methods, employing frequency-domain Newton–Raphson schemes to obtain periodic solutions in toroidal pipe geometries (Lupi et al., 2023).

Characterization is enabled by spatiotemporal and polarization state tomography, reconstructing density matrices from system projections to measure fidelity, concurrence, and entanglement of formation. For the flying doughnut pulse, concurrence in the space–time domain reaches up to 0.91, confirming near-maximal nonseparability; in space–polarization, concurrence values of $\sim$0.80 are reported (Vignjevic et al., 13 Sep 2025, Shen et al., 2020). Hyperspectral imaging, spatially resolved interferometry, and direct field mapping further quantify propagation invariance and topological invariants.

Switchable control over skyrmion modes (electric vs magnetic) is demonstrated using nonlinear metasurfaces in concert with tunable pump polarization, providing external and dynamic manipulation capability (Niu et al., 29 Aug 2025).

5. Light–Matter Interaction and Physical Realizations

Toroidal pulses interact with matter in ways not accessible to standard transverse electromagnetic waves. Their doughnut-shaped topology—with strong longitudinal components—facilitates the excitation of dynamic toroidal dipole and anapole modes in nanostructures or dielectric particles, enabling both far-field and non-radiating localized responses (Raybould et al., 2015, Zdagkas et al., 2021).

In molecular systems, pulsed microwave fields can coherently manipulate quantum toroidal moments in lanthanide complexes, enabling magnetoelectric coupling while providing resilience against stray magnetic fields (Hymas et al., 11 Apr 2025). Ultrafast control over nanoscopic toroidal moments (field-free vector potentials) is achieved with specifically tailored THz pulse sequences in nanodonut structures, providing contactless phase manipulation relevant for quantum information and magnetic device operation (Wätzel et al., 2018).

In hydrodynamic contexts, the periodic driving of toroidal pipe flow is used to investigate time-periodic base states and transitions among quasi-steady, intermediate, and plug-flow regimes as a function of oscillation frequency (Lupi et al., 2023).

6. Applications: Information, Positioning, and Sensing

The robust space–time and topological features of toroidal pulses enable several advanced applications:

• Telecommunications and Multiplexing: Encoding data in space–time or space–polarization nonseparable states enhances channel capacity and enables topologically protected, resilient transmission (Papasimakis et al., 2017, Vignjevic et al., 13 Sep 2025, Niu et al., 29 Aug 2025).
• Super-Resolution Positioning and Imaging: The nonseparable structure and skyrmion topology enable centimeter-level 3D positioning using a single antenna in free space, surpassing prior wavelength-limited resolution (Wang et al., 7 May 2024).
• Topological Information Transfer: The invariance and robustness of skyrmion or double-helix singularities under propagation allow for multidimensional encoding, low-error rate transmission, and robust data transfer, even in turbulent environments (Shi et al., 20 Sep 2024, Wang et al., 20 Aug 2024, Shen et al., 2022).
• Microscopy and Metrology: Propagation-invariant, strongly longitudinally polarized pulses (SLPTPs) provide enhanced superresolution capability, as well as access to new regimes of light–matter coupling (Wang et al., 13 May 2024).
• Quantum Technologies and Spintronics: The manipulation and readout of quantum toroidal moments in molecular complexes suggest avenues for room-temperature, long-lived qubit realization and high-density data storage (Hymas et al., 11 Apr 2025).

7. Outlook and Future Directions

Ongoing research addresses:

• Refinement of experimental generation schemes for highly pure and tunable toroidal, supertoroidal, and hybrid pulses in various spectral domains (Wang et al., 14 Jan 2025, Wang et al., 20 Aug 2024).
• Application of nonlinear metasurfaces for switchable, dynamic control of toroidal topology and skyrmion mode switching (Niu et al., 29 Aug 2025).
• Theoretical exploration of higher-order, chiral, and nontoroidal generalizations—including helical pulses with double-helix singularities and their associated vortex-antivortex dynamics (Shi et al., 20 Sep 2024).
• Quantum state tomography and advanced density matrix techniques for complete characterization and optimal use in quantum information protocols (Vignjevic et al., 13 Sep 2025, Shen et al., 2020).
• Integration of toroidal and skyrmionic pulses into classical and quantum communication platforms, metrological schemes, and advanced optical instrument design.

In summary, toroidal pulses constitute a versatile and rapidly developing frontier at the intersection of topology, ultrafast optics, electromagnetism, and quantum control, with demonstrated and anticipated impact on fundamental science and emerging information technologies.