Mesh of Spatiotemporal Optical Vortices

• M-STOVs are engineered arrays of spatiotemporal phase singularities with quantized topological charges and programmable intensity nulls.
• They are generated using techniques like phase-velocity shearing, programmable spatiotemporal holography, and perfect STOV collisions to control vortex properties.
• Experimental demonstrations from THz to visible regimes highlight their applications in optical trapping, communications, and topological photonics.

A mesh of spatiotemporal optical vortices (M-STOV) is an engineered array or lattice of phase singularities in both space and time, characterized by transverse orbital angular momentum (T-OAM) and programmable intensity nulls. These structures fundamentally generalize classic optical vortices by introducing fully spatiotemporal phase singularities and quantized topological charges in both transverse space and local time, enabling new regimes of light-matter interaction, high-dimensional information encoding, and structured light manipulation. M-STOVs can be produced and controlled through several mechanisms, including phase-velocity shearing, programmable spatiotemporal holography, and controlled collision of “perfect” Bessel-Gaussian-based vortices, and have been demonstrated experimentally from the terahertz down to the visible regime (Adams et al., 14 Nov 2025, Wu et al., 22 Nov 2025, Fan et al., 20 Jan 2025).

1. Theoretical Principles of M-STOV Formation

The formation of M-STOVs is governed by the interplay of spatial and temporal phase structuring in pulsed electromagnetic fields. The mathematically rigorous condition is that around any circuit $C$ in the $(x,t)$ or $(y,t)$ plane enclosing a singularity, the total phase $\phi(x,t)$ must wind by $2\pi m$, where $m$ is the topological charge:

$$m = \frac{1}{2\pi} \oint_C d\phi = \frac{1}{2\pi} \oint_C [\partial_x \phi\,dx + \partial_t \phi\,dt]$$

For a sharp phase-velocity boundary, the phase jump required for a unit-charge (single $\pm1$) STOV is set by the relation $\Delta \phi_0 = \omega_0 \Delta t \gtrsim \pi$, where $\omega_0$ is the carrier frequency and $\Delta t$ the local time delay across the boundary. This time delay must be at least of order of the pulse duration $\tau_p/2$ (Adams et al., 14 Nov 2025).

In design frameworks based on the spatiotemporal Fourier transform of Bessel-Gaussian beams, the M-STOV mesh is realized by superposing multiple “perfect” STOVs:

$$E_{\text{mesh}}(r, \varphi, t) = \sum_{j=1}^{N} E_{0,j} \exp\left[-\frac{(r_j - r_{0,j})^2}{w_j^2}\right] J_{\ell_j}(k_{r,j} r_j) \exp\big[i \ell_j \varphi_j + i (k_{x,j} x + k_{t,j} t + \theta_{0,j})\big]$$

Here, every constituent vortex can be individually positioned and assigned topological charge $\ell_j$, with the mesh properties determined collectively by the superposition (Fan et al., 20 Jan 2025).

2. Fabrication and Generation Methodologies

2.1 Phase-Velocity Shearing

The foundational experimental approach for M-STOV generation leverages phase-velocity contrasts across sharp boundaries in a mask or slab. By creating a binary thickness map $L(x,y)$ with values $0$ or $L_0$, one engineers a spatially varying time delay:

$\Delta t(x, y) = \frac{[ n - 1 ]\, L(x, y)}{c}$

This scalar map imprints local spatiotemporal phase jumps, and by arranging the boundaries of $L(x, y)$ in arbitrary geometries—including lines, rings, polygons, or custom 2D patterns—a corresponding mesh of STOVs is formed. Example implementations employed polylactic acid (PLA) masks fabricated on commercial FDM 3D printers for terahertz light, with typical thicknesses $L_0 \sim 0.8\,\text{mm}$ tuned for the appropriate $\Delta t \approx T_0/2$ (Adams et al., 14 Nov 2025).

2.2 Programmable Spatiotemporal Holography

An alternative methodology involves spatiotemporal pulse shaping with a folded 4f architecture and a reflective phase-only SLM. The SLM is programmed with a composite phase mask that superposes $N$ singularities, with direct mapping from lateral SLM coordinates to temporal and spatial offsets in the output field:

$$\Phi(x, y) = \sum_{j=1}^{N} \mathrm{sgn}(\ell_j)\, \tan^{-1}\left( \frac{y + d y_j}{x + d x_j} \right)$$

This directly enables precise placement, topological charge, and even “off” or “on” control of each STOV pin, yielding a fully flexible 2D mesh with controllable intensity nulls (Wu et al., 22 Nov 2025).

2.3 “Perfect” STOV Lattices

Another realization employs the coherent collision of multiple “perfect” STOV wavepackets, each constructed via the inverse spatiotemporal Fourier transform of a polychromatic Bessel–Gaussian spectrum:

$$\tilde{A}_\ell(\rho, \varphi) = \exp\left[ - \left(\frac{\omega}{\Omega_0}\right)^2 \right] J_\ell(a \rho) e^{-i \ell \varphi}$$

with the field in real $(x, T)$ given by a ring-shaped Bessel profile whose spatial and temporal diameters are independent of $\ell$. Arranging multiple such packets at prescribed intersection points yields a periodically or aperiodically structured M-STOV, with mesh periodicities $\Lambda_x$, $\Lambda_t$ set by beam tilt differences (Fan et al., 20 Jan 2025).

3. Topological Structure and Control in M-STOV

Each node (or “pin”) of the M-STOV represents a quantized phase singularity with integer or half-integer charge, as determined by the phase jump magnitude. The total topological charge of the mesh is algebraic (i.e., sum of all node charges). By spatially arranging pins with controlled sign (helicity) and charge, complex reconnection dynamics and the emergence of higher-order topological features can be systematically engineered (Adams et al., 14 Nov 2025, Wu et al., 22 Nov 2025).

Programmable phase-contrast techniques allow real-time control: switching node charges, spatial locations, or even suppressing selected singularities. The mesh topology may thus be dynamically reconfigured, presenting a high-dimensional optical platform for OAM-based information encoding and topological photonics studies.

4. Experimental Realizations and Diagnostics

4.1 THz M-STOV Demonstrations

Demonstrations at 0.3 THz employed single-cycle pulses shaped with PLA masks. Mask geometries included straight edges (line vortices), rings, squares, triangles, and arbitrarily curved features. The resulting M-STOVs were diagnosed by raster-scanned electro-optic sampling, revealing the progression and reconnection of multiple topological singularities over propagation distance (Adams et al., 14 Nov 2025).

4.2 Optical and Near-IR M-STOV Experiments

In the visible and near-infrared, M-STOVs were generated via SLM-based pulse shaping of 35 fs pulses at 800 nm. The spatial and temporal structure was mapped by imaging the output onto a camera after space–spectrum mixing (using a 4f system and cylindrical lenses), providing direct measurement of the spatial–spectral decomposition of the vortex mesh. Intensity nulls and OAM values were unambiguously identified from the number and arrangement of Hermite–Gaussian lobes and phase-slip gaps in the spatial–spectral domain (Wu et al., 22 Nov 2025).

A tabular summary of key implementation regimes:

Frequency Regime	Fabrication Method	Typical Feature Size
THz (0.3 THz)	3D-printed PLA mask	$L_0 \sim 0.8$ mm
Visible/IR (800 nm)	SLM pulse shaping	$L \sim 7.5-150\,\mu$m
Both	ST Fourier holography	ring $r_0$ set by optics

5. Nonlinear and Advanced Manipulation of M-STOVs

M-STOV fields exhibit robust nonlinear optical effects. Second-harmonic generation (SHG) in a thin BBO crystal produces a new M-STOV configuration where each local charge doubles ($\ell_j \to 2\ell_j$), and the spectrum narrows by $\sqrt{2}$ due to nonlinear phase-matching effects:

$$E_{2\omega}(y,t) \propto \left[E_\omega(y,t)\right]^2$$

The number of Hermite–Gaussian lobes measured in far-field diffraction patterns verifies the OAM doubling and reveals the increased dimensionality of the frequency representation of the mesh (Wu et al., 22 Nov 2025).

6. Applications and Prospects

The ability to synthesize and program arbitrary M-STOV geometries enables several advanced applications:

• Ultrafast optical trapping with multiple independently addressable spatiotemporal traps (“optical pins”) (Wu et al., 22 Nov 2025).
• High-capacity optical communications via OAM multiplexing in spatiotemporal degrees, achieving complexity beyond conventional vortex beams (Wu et al., 22 Nov 2025, Fan et al., 20 Jan 2025).
• Analog optical computing through Hermite–Gaussian field multiplexing and programmable trap lattices.
• Photonic topology: model systems for higher-dimensional topological defects, exotic reconnection behavior, and experimental analogues of condensed-matter vortices (Adams et al., 14 Nov 2025).
• Ultrafast microscopy and manipulation: time-gated arrays of vortex spots for imaging and micro-fabrication (Fan et al., 20 Jan 2025).

Extensions to mid-infrared and visible frequencies are enabled by established micro-machining techniques, enabling M-STOV structures at the micron scale (Adams et al., 14 Nov 2025).

7. Scaling Laws, Limitations, and Future Directions

The minimum local time delay for optimal STOV formation is set by the relation $L \approx c\,\tau_p / [2(n-1)]$. In broadband regimes, the required mask or SLM structure must account for material dispersion, with $L(x,y)$ adapted so that the spectrally averaged delay matches the pulse envelope width. The propagation stability of “perfect” STOVs is limited by Rayleigh length and dispersion, but phase-locked designs and digital holography enable robust mesh creation. A plausible implication is that future research will focus on integrating M-STOV modules within reconfigurable photonic circuits and continuous-time quantum information networks (Wu et al., 22 Nov 2025, Fan et al., 20 Jan 2025).

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References:

• (Adams et al., 14 Nov 2025) Arbitrary geometry electromagnetic spatiotemporal vortices from phase velocity shearing
• (Wu et al., 22 Nov 2025) Mesh of Spatiotemporal Optical Vortices with Programmable Intensity Nulls
• (Fan et al., 20 Jan 2025) Perfect Spatiotemporal Optical Vortices