Nondiffracting supertoroidal pulses and optical “Kármán vortex streets”

The study of topological properties of light has revealed profound implications for light–matter interactions, nonlinear optics, spin–orbit coupling, microscopy, metrology, and information transfer. Structuring light in space and time enables three-dimensional topological field configurations such as toroidal phase vortices, photonic skyrmions, and hopfions. Toroidal light pulses, notable for space–time nonseparability and isodiffraction, can engage toroidal excitations in matter. Supertoroidal pulses (STPs) generalize toroidal pulses and exhibit fractal-like singularities, vortex rings, energy backflow, and skyrmionic patterns. However, previously known skyrmionic light structures and STP topologies are short-lived and do not persist upon propagation. This work addresses the gap by introducing nondiffracting supertoroidal pulses (NDSTPs) that preserve skyrmion-like and vortex configurations over long distances, aiming to realize robust propagation of complex topological electromagnetic structures and explore their analogies to Kármán vortex streets in fluid dynamics.

Prior work established space–time structured pulses and particle-like topologies in light, including toroidal pulses and photonic skyrmions/hopfions. STPs were shown to possess complex topology with fractal-like singularities and energy backflow but suffered from rapid evolution and loss upon propagation. Classical nondiffracting wave solutions (e.g., Bessel beams, Bessel-X pulses) demonstrate propagation invariance but are typically scalar, long-pulse, or idealized infinite-energy constructs, lacking the few-cycle, vectorial toroidal topology. Optical analogs of vortex streets were reported in stationary or nonlinear regimes, whereas robust propagating pulse analogs remained unexplored. The electromagnetic directed-energy pulse trains (EDEPT) framework provides localized, finite-energy, space–time nonseparable solutions, and prior toroidal pulse solutions imposed q3 → ∞, leaving finite-q3 regimes unexplored. This work builds on these foundations to realize few-cycle, vectorial NDSTPs with persistent topological properties.

• Theoretical framework: Start from the EDEPT theory to construct localized, finite-energy, space–time nonseparable solutions of Maxwell’s equations. A scalar generating function f(r,t) satisfies the scalar wave equation ∇²f − (1/c²) ∂²f/∂t² = 0 in free space.
• Modified power spectrum solution: Use Ziolkowski’s modified power spectrum method to obtain f(r,t) = f0 e^(−s/q3) / [(q1 + iτ)(s + q2)], where s = r²/(q1 + iτ) − iσ, τ = z − ct, σ = z + ct. Parameters q1, q2, q3 > 0 (length units) and α ≥ 1 (dimensionless). α controls energy confinement/divergence; α = 1 ensures finite energy and minimal divergence.
• Field construction: Build the vector Hertz potential Π = ∇ × ∇ × f(r,t). TE-mode fields are obtained from E = −με ∂t (∇ × Π) and H = ∇ × (∇ × Π). TM mode follows by exchanging E and H.
• Role of parameters: For prior STPs (q3 → ∞), q1 sets effective wavelength (q1 = 0.24λ), q2 relates to Rayleigh length (q2 = z0/2). Introducing finite q3 quantifies transverse divergence at focus and suppresses longitudinal divergence, leading to nondiffracting behavior. Decreasing q3 squeezes the envelope (dumbbell to X-shaped) and, for q3 ≤ q1 with α = 1 and q1 < q2, yields NDSTPs.
• Propagation analysis: Numerically evaluate transverse full width at half maximum (FWHM) versus z for q2 = 100 q1 and varying q3 from ∞ to q1 over z up to 10 q1 to assess diffraction. Spatiotemporal field evolution is visualized for multiple q3 values, revealing conical/X-type structures for q3 ≈ q1.
• Topological analysis: Compute vector-field singularities (vortices, saddles) in H(r,t), identify skyrmionic textures on transverse planes, and evaluate deep-subwavelength reversal regions (FWHM scales ~ q1/2 and q2/10). Analyze Poynting vector S for forward/backflow layering and source–sink behavior mediated by vortex arrays.
• Spectral analysis: Calculate plane-wave spectra and ω–k projections. Apply nondiffraction criterion for space–time wave packets: NDSTP spectrum confined to a conical section (plane cut of light cone) with near-constant group velocity v ≈ c; quantify forward-propagating energy fraction.
• Practical generation considerations: Discuss shaping ω–k spectra (thin conic line with central null at k⊥ = 0) using photonic crystal slabs (PCS) with controlled geometry/symmetry for cylindrically polarized emission, or transformation-optics-based approaches; assess aperture-size requirements via numerical study.

• Nondiffracting supertoroidal pulses (NDSTPs): For q1 < q2, q3 = q1, α = 1, pulses propagate with negligible diffraction, maintaining complex toroidal vector topology over very long distances.
• Diffraction control via q3: Reducing q3 from ∞ weakens divergence; near-nondiffracting behavior achieved for q3 ≤ 5 q1, and fully NDSTP regime for q3 = q1 (α = 1). Pulse becomes X-shaped with conical evolution yet preserves toroidal topology.
• Robust topological structures: NDSTPs exhibit propagation-robust fractal-like singularity patterns and matryoshka-like shells where E vanishes, persisting over arbitrarily long distances rather than only near focus as in STPs.
• Skyrmions and singularities: Magnetic field shows off-axis vortex and on-axis saddle singularities creating multiple electromagnetic skyrmions in transverse planes. Deep-subwavelength vector reversal regions occur on scales ~ q1/2 and q2/10. Skyrmion textures persist and alternate periodically among four types (two polarities × two helical angles); observable skyrmion numbers are ±1 in transverse cuts with total skyrmion number zero by symmetry.
• Energy flow: Poynting vector reveals layered forward-flow and backflow; front-half vortices act as energy sources and rear-half as sinks with extended backflow regions between them, while high-intensity regions support net forward transport.
• Optical Kármán vortex street analogy: NDSTPs form staggered, propagating arrays of vortex dipoles (vortex-ring streets in 3D due to cylindrical symmetry), realizing a linear optical analogue of von Kármán vortex streets.
• Spectral evidence of nondiffraction: NDSTP spectra lie on a thin conic section plane on the light cone with near-constant group velocity v ≈ c; over 99% of pulse energy is in forward-propagating waves.
• Practical propagation length: Numerical study indicates a metasurface/aperture of ~400 λ yields nondiffracting propagation (FWHM variation within 1%) over ~10^5 λ.
• Finite energy advantage: Unlike ideal Bessel beams, NDSTPs are finite-energy solutions, suggesting reduced sensitivity to finite aperture and closer adherence to ideal behavior in practice.

The work addresses the central challenge of realizing electromagnetic pulses that retain complex, skyrmion-rich topology upon propagation. By introducing q3 as a transverse-divergence control parameter and identifying the NDSTP regime (q3 = q1, α = 1, q1 < q2), the authors achieve near-nondiffracting propagation while preserving STP topological features. The spectral confinement on a conic section of the light cone explains the uniform group velocity and absence of diffraction, while maintaining finite energy and cylindrical vector structure. The persistent skyrmionic textures, fractal-like singularity shells, and layered energy backflow establish NDSTPs as robust carriers of topological information. The observed staggered vortex-ring streets provide an optical analogue of Kármán vortex streets in a linear, propagating pulse context, suggesting new avenues to study transport analogies between fluids and structured light. Practical generation routes are discussed via photonic crystal slabs capable of tailoring ω–k distributions and vector polarization, and transformation-optics approaches extended to broadband, vectorial pulses. Compared to Bessel beams, the finite-energy nature of NDSTPs reduces sensitivity to aperture size, supporting realistic implementations over long distances for applications in communications, remote sensing, and metrology.

The study introduces nondiffracting supertoroidal pulses (NDSTPs) that maintain robust topological features—fractal-like singularities, skyrmions, vortex rings, and energy backflow—over arbitrarily long propagation distances. NDSTPs realize a linear optical analogue of Kármán vortex streets via staggered vortex-ring arrays and provide a platform to investigate propagation dynamics of electromagnetic skyrmions and their interactions with complex media. Spectral analysis confirms near-uniform group velocity with >99% forward energy, and practical modeling shows that a ~400 λ aperture supports nondiffracting behavior over ~10^5 λ. The persistence of topology suggests encoding information in topological features for long-distance transfer and applications in telecommunications, LIDAR, toroidal spectroscopy, and precision metrology. Future work includes experimental generation via photonic crystal slabs or transformation optics, optimization of ω–k engineering for broadband, cylindrically polarized emission, and studies in nonlinear, chiral, or anisotropic media.

• Nondiffraction is approximate: exact nondiffraction requires an infinitely thin spectral line; NDSTPs closely approximate this but remain finite-bandwidth, finite-energy wave packets.
• Generation challenges: Two key requirements must be met experimentally—engineering a thin conic-section ω–k spectrum and enforcing a central null at k⊥ = 0 due to cylindrical polarization and vector singularity structure.
• Aperture constraints: Although less sensitive than Bessel beams, achieving long nondiffracting distances still benefits from large apertures (e.g., ~400 λ for ~10^5 λ propagation in simulations).
• Assumptions: Analysis assumes free-space propagation; behavior in complex media may introduce dispersion or nonlinear effects requiring further study.
• Parameter sensitivity: Precise control of q1, q2, q3 and α is needed to access the NDSTP regime and maintain topology over distance.