Vector & Topological Airy Beams

• The paper demonstrates robust methods for synthesizing vector Airy beams with structured polarization using geometric-phase elements and dual SLM systems.
• Key experiments reveal self-healing and accelerating properties along with topological features such as vortex singularities and skyrmion lattices.
• The study highlights practical applications in optical trapping, high-power material processing, and advanced microscopy through propagation invariance and spatial nonseparability.

Vector and topological Airy beams constitute a broad class of structured electromagnetic fields distinguished by their accelerating, self-healing propagation and their finely structured polarization or topological attributes. Scalar Airy beams, as non-diffracting solutions to the paraxial wave equation, have been extensively studied for their robust main lobe and self-acceleration. The extension to vector beams—fields with spatially variant polarization—opens access to joint control over amplitude, phase, polarization, and topological defects. Recent advances have demonstrated high-power, pulsed vector Airy beams with radial and azimuthal polarization; the systematic design and analysis of abruptly autofocusing and vortex-carrying Airy-type beams; and the characterization of richly detailed polarization textures including skyrmions and antiskyrmions in both theory and experiment (Berškys et al., 13 Jan 2025, Zhao et al., 2021, Hu et al., 2022, Suzuki et al., 2021).

1. Mathematical Foundations of Airy and Vector Airy Beams

The scalar (1+1)-dimensional Airy beam is a solution to the paraxial wave equation: $$\frac{\partial u}{\partial z} = \frac{i}{2k} \frac{\partial^2 u}{\partial x^2}$$ with finite-energy regularization,

$u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$

where $s = x/x_0$, $\xi = z/(k x_0^2)$, $a$ is the truncation parameter, and $x_0$ the lateral scale (Berškys et al., 13 Jan 2025).

Extension to two transverse dimensions generalizes the solution by taking products over axes. The angular spectrum representation factorizes as $G(k_x, k_y) = g_x(k_x) g_y(k_y)$, with each factor incorporating a cubic phase, which imparts the signature Airy acceleration.

For vectorial generalizations, two orthogonal solenoidal modes are constructed: $$\mathbf M(\mathbf{r}) = \nabla \times (\hat{\mathbf e}_z U), \qquad \mathbf N(\mathbf{r}) = \frac{1}{k} \nabla \times \mathbf M(\mathbf{r}),$$ where $U$ is the scalar envelope. $\mathbf{M}$ exhibits purely azimuthal transverse polarization; $u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$0 is predominantly radial plus longitudinal. Explicitly,

$u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$1

and, in the paraxial limit at $u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$2,

$u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$3

which constitutes the canonical azimuthal polarization profile for Airy envelopes. Radial vector beams are similarly constructed (Berškys et al., 13 Jan 2025).

More generally, vector Airy beams can be formulated as superpositions of scalar Airy or Airy–type modes in orthogonal polarization bases, giving rise to spatial–polarization nonseparability (Zhao et al., 2021, Hu et al., 2022).

2. Vectorial Airy Beams: Construction and Polarization Structure

Vector Airy beams are constructed through superpositions of spatially structured scalar Airy or related beams, each encoded in an orthogonally polarized field. The general form for a parabolic-accelerating vector beam is

$u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$4

where $u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$5 and $u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$6 are parabolic-Airy scalar modes, and $u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$7 span an arbitrary elliptical polarization basis (Zhao et al., 2021).

Circular Airy Gaussian vortex beams, used to realize abruptly autofocusing vector Airy beams, are given by (Hu et al., 2022): $u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$8 with vectorialization obtained by,

$u(x,z) = \Ai \bigl[ s - \tfrac{1}{2} \xi^2 + i a \xi \bigr] \exp \Big\{ a (s - \tfrac{1}{2}\xi^2) + \tfrac{i}{2} \xi (a^2 + s - \frac{\xi^2}{3}) \Big\},$9

Vector Airy beams thus possess a highly tunable, non-uniform polarization state across the transverse plane. Experimentally reconstructed Stokes vector fields demonstrate that the local polarizations trace out a single great circle on the Poincaré sphere (constant azimuth, varying ellipticity), fully controlled by the amplitudes and relative phase of the superposed modes (Zhao et al., 2021).

3. Topological and Polarization Singularities

Scalar Airy beams lack orbital angular momentum (OAM) singularities. The introduction of vectorial structure or embedded vortex modes fundamentally alters their topological character. For example:

• Parabolic-accelerating vector waves constructed from superpositions without net azimuthal phase produce no integer OAM, but the local polarization map features robust C-points (points of pure circular polarization) and L-lines (lines of linear polarization). These are mapped onto a great circle of the Poincaré sphere and are propagation invariant up to the underlying parabolic shift (Zhao et al., 2021).
• Circular Airy Gaussian vortex vector beams systematically distribute polarization singularities, depending on the relative charges $s = x/x_0$0; these can be characterized by interferometric or Stokes polarimetry measurements. The intermodal phase rotates with propagation, and such beams inhabit higher-order Poincaré spheres (Hu et al., 2022).
• In high-power ultrafast vector Airy beams, skyrmion and antiskyrmion lattice textures can be realized in the polarization field. These are topologically protected, particle-like structures within the Stokes parameter distribution, confirmed in both simulation and experiment (Berškys et al., 13 Jan 2025).

4. Experimental Generation and Characterization Techniques

A range of methods enable the synthesis of vector and topological Airy beams:

• Geometric-phase elements (GPEs): Direct-write femtosecond laser inscription in fused silica is employed to create nanograting-based GPEs, enacting spatially tailored birefringence for the robust and efficient generation of radially and azimuthally polarized Airy beams at high pulse energies. The orientation of the fast axis and the retardance (approximate $s = x/x_0$1) determine the imposed geometric phase (Berškys et al., 13 Jan 2025).
• Spatial Light Modulators (SLMs): For abruptly autofocusing vector Airy beams and CAGV beams, dual SLM holography separates horizontally and vertically polarized components, each encoding a distinct Airy-Gaussian vortex mode before recombination and conversion to the circular basis (Hu et al., 2022).

Comprehensive Stokes polarimetry is performed to reconstruct the transverse polarization structure. This involves measuring intensities after passage through specific analyzer settings to extract $s = x/x_0$2, from which local ellipticity, orientation, and degree of polarization nonseparability (vector quality factor, VQF) are derived (Berškys et al., 13 Jan 2025, Zhao et al., 2021, Hu et al., 2022).

5. Airy Beams with Embedded Vortices and Propagation Invariance

Embedding vortices in the Airy beam profile yields beams whose main intensity lobe and topological singularity co-move along the Airy trajectory. A key advance is the synthesis of a propagation-invariant vortex Airy beam by superposing two laterally shifted Airy beams with a $s = x/x_0$3 phase offset: $s = x/x_0$4 where $s = x/x_0$5 and $s = x/x_0$6 are the first zeros of the Airy function and its derivative, respectively. This construction yields a topological charge $s = x/x_0$7 singularity that is rigidly locked to the self-accelerating Airy main lobe (Suzuki et al., 2021).

Quantifying the OAM purity reveals a significant gain in single-mode occupancy over conventional vortex Airy beams (e.g., 91% for the propagation-invariant type vs. 81% for the conventional type) (Suzuki et al., 2021).

6. Applications and Future Directions

Vector and topological Airy beams exhibit several properties with direct relevance to advanced photonic applications:

• Optical Trapping and Manipulation: The spatially varying polarization exerts spin-dependent optical forces, enabling finely configurable micro- and nanoparticle manipulation and controlled spinning (Zhao et al., 2021, Berškys et al., 13 Jan 2025).
• High-Power Material Processing: Robust, high-power, ultrafast Airy vector beams enable alignment of intense, curved focal lines with engineered polarization for precise laser machining, waveguide writing, and the fabrication of complex 3D photonic structures (Berškys et al., 13 Jan 2025).
• Enhanced Microscopy: Propagation-invariant vortex Airy beams benefit STED-SPIM and light-sheet microscopy by delivering a vortex-shaped depletion beam that accurately tracks the excitation lobe with minimal distortion (Suzuki et al., 2021).
• Structured Communications: The inherent spatial–polarization nonseparability and multiplexed OAM–spin structure create new channels for information encoding in free-space optical communications (Hu et al., 2022, Zhao et al., 2021).
• Nonlinear Optics: The preservation of phase topology over long distances is expected to improve phase matching and energy transfer in nonlinear generation schemes (Suzuki et al., 2021).

7. Perspectives and Theoretical Implications

Vector and topological Airy beams provide a unique platform for studying light's geometric, topological, and dynamical aspects in highly controlled experiments. The robust generation of skyrmion and antiskyrmion lattices in the polarization field paves the way for systematic photonic studies of topological defects in analogy with condensed matter systems (Berškys et al., 13 Jan 2025). The distinction between structural topological charge (OAM) and polarization singularities (C-points, L-lines, skyrmions) is crucial for understanding their field evolution, stability, and potential for engineering light–matter interaction.

Tunable parameters, such as amplitude ratios and phase offsets in vector superpositions, provide full control over the polarization texture’s topology and geometry. This continuous mapping between scalar and vector states, and between homogeneous and topologically nontrivial polarization patterns, marks a versatile frontier in structured light research.

References:

(Berškys et al., 13 Jan 2025, Zhao et al., 2021, Hu et al., 2022, Suzuki et al., 2021)