Airy Beams: Properties and Applications

• Airy beams are non-diffracting, self-accelerating wave packets defined by the Airy function, exhibiting parabolic trajectories, shape invariance, and self-healing properties.
• Finite-energy implementations via truncation and apodization modify the ideal beam profile, balancing diffraction resistance and practical generation in both free-space and on-chip systems.
• Generation techniques using cubic phase profiles and spatial light modulators enable applications in optical communications, particle manipulation, and integrated photonics.

Airy beams are non-diffracting, self-accelerating wave packets described by the Airy function, known for their parabolic trajectory and self-healing properties. As exact solutions to the paraxial wave equation, Airy beams have found applications across optics, matter wave physics, near-field communications, photonic integration, and structured beam engineering. Their realizable forms require finite-energy modifications, leading to the rich variety of propagation phenomena, practical generation methods, and fundamental physical constraints detailed below.

1. Fundamental Properties and Mathematical Formulation

Airy beams in their idealized form are infinite-energy solutions to the paraxial wave equation, first theoretically analyzed by Berry and Balazs as non-spreading, self-accelerating wave packets. In the optical context, the one-dimensional paraxial field is given by: $$\psi(x, z) = \exp\left\{i \left[ xz - \frac{z^3}{12} \right]\right\} \operatorname{Ai}\left(x - \frac{z^2}{4}\right)$$ where $\operatorname{Ai}$ denotes the Airy function.

Key properties include:

• Shape invariance: Intensity profiles remain unchanged under free-space propagation, modulo a transverse parabolic shift.
• Self-acceleration: The beam's main lobe follows a curved (parabolic) transverse trajectory: $x(z) = x(0) + z^2/4$.
• Non-diffraction: Ideal Airy beams maintain their profile indefinitely (neglecting truncation).
• Self-healing: The field reconstructs its main lobe after partial obstruction, owed to energy flow from the secondary lobes.

These features arise from the Airy function’s connection to the solution of the linear potential in the Schrödinger equation and are extended to two transverse dimensions as separable products of one-dimensional Airy functions, or to more general forms by engineering the input phase profile (Rogel-Salazar et al., 2014, Efremidis et al., 2019).

2. Finite-Energy Airy Beams and Practical Implementation

Physical realization requires finite energy, necessitating truncation by apodization (e.g., exponential or Gaussian), finite apertures, or both (Zamboni-Rached et al., 2012, Sanz et al., 2023, Darsena et al., 19 Aug 2025). This modification leads to distinctive propagation phenomena:

• Limited diffraction-free range: After a critical propagation distance (determined by the truncation parameters), the beam spreads and loses its ideal profile (Rogel-Salazar et al., 2014).
• Modified self-acceleration: Only the leading portion of the intensity profile retains the parabolic trajectory; the rear lobes disperse, and energy is transferred between lobes during propagation (Sanz, 2022, Sanz et al., 2023).
• Self-healing persists: Despite truncation, self-reconstruction features endure within the diffraction-free propagation window.

Analytic and semi-analytic formulations for truncated Airy beams have been developed, including superpositions of exponentially decaying Airy functions for rapid and accurate characterization (Zamboni-Rached et al., 2012). The impact of the truncation function—Gaussian, Lorentzian, Sinc—was shown to affect the phase and amplitude content of the beam, thereby influencing the preservation of self-accelerating trajectories (Sanz et al., 2023).

3. Generation Techniques: Free-Space, On-Chip, and Structured Matter

Free-Space Generation

Airy beams are routinely generated by imprinting a cubic phase profile on a Gaussian beam, typically using a spatial light modulator (SLM), followed by a Fourier-transform lens to produce the Airy field (Suarez et al., 2015, Latychevskaia et al., 2016). Modulating the cubic phase parameter allows control over the acceleration and trajectory. Advanced methods utilize computational and photorefractive holography for dynamic, high-resolution generation (Suarez et al., 2015), as well as transmissive SLMs with negative transmission to directly encode the Airy function (eliminating the need for a physical lens and miniaturizing the setup) (Latychevskaia et al., 2016).

Integrated Photonics

One-dimensional Airy beams have been realized on silicon photonic chips using meta-optics that convolve a cubic phase with a lens phase within a $3 \times 16~\mu\text{m}^2$ footprint. These beams exhibit self-healing, broadband operation, and extended depth-of-focus, providing a route to miniature, alignment-free, diffraction-resistant beam routing and on-chip manipulation (Fang et al., 2021).

Plasmonic and Electron Beams

Plasmonic Airy beams, generated on silver surfaces via nano-arrays with graded spacing, realize SPP (surface plasmon polariton) Airy beams that maintain subwavelength confinement and are suitable for on-chip photonics and nanoparticle manipulation (Li et al., 2011). Electron Airy beams, first demonstrated via nanofabricated holograms imposing a cubic phase, exhibit analogous self-acceleration, self-healing, and can be manipulated with electromagnetic fields, opening novel directions for electron interferometry and beam shaping (Voloch-Bloch et al., 2012).

4. Propagation Dynamics, Physical Constraints, and Flux Trajectories

While the ideal Airy beam’s shape invariance and acceleration derive from its infinite extent, practical beams are subject to quantitative limitations:

• Transverse and longitudinal extents: Finite truncation is mandated by the requirement that the lobe spacing exceed a critical value related to the optical wavelength, yielding explicit expressions for beam extent and the “region of existence” (Rogel-Salazar et al., 2014).
• Traveling wave decomposition: Airy beams can be represented as the superposition of two Hankel-like traveling waves, illuminating the self-healing and double-focusing behavior (Rogel-Salazar et al., 2014).
• Nonparaxial corrections: Full-wave analyses clarify that shape-preserving Airy beams, especially when generated from incomplete Bessel fields, may follow elliptical rather than parabolic caustics as nonparaxial corrections become significant (Zapata-Rodriguez et al., 2014).
• Flux trajectory analysis: Insight into energy flow and the evolution of finite-energy beams is gained by mapping flux trajectories; in ideal beams, lobes remain dynamically segregated, whereas in finite-energy beams, trajectory transfer among lobes leads to the transition toward a Gaussian envelope with increasing propagation (Sanz, 2022, Sanz et al., 2023).

The escape rate parameter, calculated from the confined energy in the leading peak, quantitatively characterizes the survival of self-accelerating behavior post truncation (Sanz et al., 2023).

5. Generalizations: Vector, Structured, Partially Coherent, and Topological Airy Beams

Vector and Topological Airy Beams

Extensions have enabled the realization of ultrafast radially and azimuthally polarized Airy beams, produced by combining scalar Airy solutions with vectorial curl operations and S-waveplates. These beams exhibit particle-like lattice topologies in their electric field and Stokes parameter distributions, with experimental evidence for skyrmionic and antiskyrmionic structures (Berškys et al., 13 Jan 2025). The internal polarization structure and topological field texture have significance for light–matter interaction, particle trapping, and topological photonics.

Symmetric and Shaped Airy Beams

Symmetrizing the angular spectrum (e.g., via absolute value in the cubic phase) yields Airy–Scorer beams with non-standard caustic morphologies and enhanced natural focusing, confirmed through phase engineering on SLMs (Jauregui et al., 2014). Likewise, adaptive control of input amplitude and phase enables beams that focus abruptly or propagate along prescribed convex trajectories (Efremidis et al., 2019).

Partially Coherent and Airy Bump Beams

Partially coherent Airy beams, modeled using cross-spectral density formalism, retain shape invariance and self-acceleration statistically even with substantial incoherence, provided the Airy structure is preserved. Critical propagation distances mark the range over which these features persist in finite-energy experimental beams. Airy beams on an incoherent Gaussian-correlated background, called ABIBs, are non-accelerating, with their propagation determined by the interplay between coherence width and aperture size (Hajati et al., 2021, Martínez-Herrero et al., 2022).

6. Applications: Communications, Particle Manipulation, Photonic Devices, and Resonant States

Airy beams’ ability to follow curved trajectories, resist diffraction, and heal after perturbation underpins their utility in diverse areas:

• Free-space optical communications: Airy beams can route signal energy around obstacles via tailored ballistic trajectories, resulting in up to 9 dB higher received power vs. Gaussian beams in obstruction scenarios (Zhu et al., 2018). Analytical models quantify the trade-off between finite aperture truncation, diffraction resistance, and self-healing for near-field wireless applications (Darsena et al., 19 Aug 2025).
• Optical trapping and micromanipulation: Self-bending and high-intensity lobes are exploited for guiding and sorting microparticles in both optical and plasmonic regimes; robust operation over extended depth-of-focus is advantageous for integrated manipulation (Suarez et al., 2015, Fang et al., 2021).
• Nonlinear optics and filamentation: Airy beams’ high field localization and curved propagation have been applied in generating plasma channels and abrupt autofocusing, with implications for material processing (Efremidis et al., 2019).
• Integrated photonics and plasmonic circuits: Planar confinement and the possibility of on-chip realization enable chip-scale photonic routing, switching, and energy localization (Li et al., 2011, Fang et al., 2021).
• Resonances in periodic media: Airy resonances in photonic crystal potentials, realized by linear variation in lattice constant (superpotential), map onto non-Hermitian Schrödinger problems and lead to unique spectral and radiative properties closely related to the Airy function paradigm (Zhang et al., 8 Oct 2025).
• Topological photonics: Vectorial Airy beams exhibiting skyrmion-like features open avenues for the paper and exploitation of topological field structures in structured light (Berškys et al., 13 Jan 2025).

7. Future Directions and Open Challenges

The field continues to expand with advances in ultra-broadband Airy beams free of dispersion via reflective phase elements (Valdmann et al., 2018), trajectory-adaptive beam shaping for wireless communications, and more sophisticated on-chip Airy sources. Open challenges include optimizing beam truncation to maximize propagation-invariant range, designing for resilience in turbulent or random media (including partially coherent and ABIB forms), scaling photonic integration for arbitrary trajectory engineering, and exploiting topological and vectorial features for advanced information processing and light-matter control.

The ongoing development of analytic, computational, and experimental methodologies for shaping, characterizing, and applying Airy beams is central to both fundamental wave physics and emerging photonic and quantum technologies.