Neutron Airy Beams & OAM Hybrids

• Neutron Airy beams are spatially structured neutron wavepackets with non-diffracting, self-accelerating profiles, achieved through cubic phase imprinting.
• The generation method employs nanofabricated silicon phase gratings to produce pure Airy, OAM, and hybrid states, with single-pass diffraction efficiencies around 1% per mode.
• Applications span advanced neutron scattering, quantum sensing, and materials characterization, enabling studies of topological spin textures and multimode entangled interferometry.

Neutron Airy beams are spatially structured neutron wavepackets characterized by a non-diffracting and self-accelerating transverse profile akin to the optical Airy beam, extending the domain of orbital angular momentum (OAM) beams in neutron science to include translation-invariant but non-paraxial solutions. The first experimental realizations of neutron Airy beams and OAM–Airy hybrid states, achieved via nanofabricated phase-grating techniques, furnish new degrees of freedom for neutron scattering, materials characterization, and quantum foundations. With the advent of coherent superpositions and multimode multiplexing of OAM and Airy states, Airy neutron beams now constitute an integral component of next-generation structured neutron sources for advanced quantum and topological matter studies.

1. Theoretical Foundations of Neutron Airy Beams

The prototypical Airy beam solution in free space is obtained by applying a cubic phase to a monochromatic plane wave. The resulting ideal wavefunction in Cartesian coordinates, for a neutron of wavenumber $k$, is:

$$\psi_\mathrm{Ai}(x, y, z) \propto \mathrm{Ai}\left[ \frac{x}{x_0} - \frac{z^2}{4k^2 x_0^4} \right]\, \mathrm{Ai}\left[ \frac{y}{y_0} - \frac{z^2}{4k^2 y_0^4} \right] \ \times \exp \left[ i \left( \frac{x z}{2 k x_0^3} + \frac{z^3}{12 k^3 x_0^6} \right) \right] \exp \left[ i \left( \frac{y z}{2 k y_0^3} + \frac{z^3}{12 k^3 y_0^6} \right) \right],$$

where $x_0, y_0$ set the characteristic transverse scales and $\mathrm{Ai}$ denotes the Airy function. This form imparts the well-known nondiffracting and transverse "self-accelerating" behavior of the Airy beam in the $x$ and $y$ directions. Experimental generation requires truncation to a finite transverse extent.

Airy beams are distinct from OAM-carrying Laguerre–Gaussian (LG) modes, which are characterized in cylindrical coordinates by an azimuthal phase factor $\exp(i\ell\phi)$ and radial structure. Their hybridization enables an expanded class of fundamental neutron wavepackets (Lailey et al., 10 Nov 2025).

2. State Generation via Phase-Grating Techniques

Neutron Airy beams are experimentally realized through nanofabricated silicon phase gratings imprinted with specific phase profiles. The cubic (Airy) phase is implemented as:

$\phi_\mathrm{cub}(x, y) = c_x x^3 - c_y y^3,$

with $c_x, c_y$ dictating the Airy beam transverse scale. Practically, this is realized by etching a "kinoform" grating with height profile proportional to $\phi_\mathrm{cub}(x, y)$, resulting in a transmission function:

$$G_\mathrm{cub}(x, y) = \frac{1}{2} \, \mathrm{sign}\left[ \sin\left( \frac{2\pi x}{p} + c_x x^3 - c_y y^3 \right) \right] + \frac{1}{2},$$

where $p$ is the grating period (typically $p=120\,\mathrm{nm}$) (Lailey et al., 10 Nov 2025).

By stacking multiple gratings—fork-dislocation (OAM) and cubic (Airy) types—along the neutron beam axis, intricate coherent superpositions can be synthesized, including pure OAM, pure Airy, and OAM–Airy hybrid states. Each grating diffracts into a well-defined first-order ($m=+1$) spatial mode, with the output wavefunction at the detector given by:

$$\Psi(\mathbf{r};z) = \sum_{j=1}^N A_j \psi_{p_j,\ell_j}(\mathbf{r};z) + \sum_{k=1}^M B_k \psi_{\mathrm{Ai},k}(x,y;z),$$

where $A_j, B_k$ are determined by relative phase-grating heights and separations, and the $\psi_{p_j,\ell_j}$ are LG (OAM) and $\psi_{\mathrm{Ai},k}$ Airy modes, respectively.

3. Experimental Realization and Characterization

The key apparatus comprises several 0.5 cm × 0.5 cm silicon wafers, each patterned with $6.25 \times 10^6$ $1\,\mu \mathrm{m} \times 1\,\mu \mathrm{m}$ kinoforms at $p=120\,\mathrm{nm}$ pitch. Two wafers are fork gratings ($q=3$, $q=7$, with etch depths 500 nm and 400 nm) and one is a cubic grating (300 nm etch). Stacking is achieved with wafer spacings of $d_1 \approx 3.3$–$3.5$ mm (Lailey et al., 10 Nov 2025).

Experiments at the GP-SANS instrument (HFIR, ORNL) use a source aperture ($\lambda_0 \approx 12\,\mathrm{\AA}$, $\Delta \lambda/\lambda \approx 13\%$), followed by a sample aperture preceding the grating stack. The detection plane sits $15.4$ m downstream, with a position-sensitive detector ($1\,\mathrm{m}^2$, pixel size $5.5\,\mathrm{mm} \times 4.3\,\mathrm{mm}$).

Measured results include:

• Just Airy beam: Single cubic grating yields a set of characteristic main and secondary Airy lobes at the detector.
• Hybrid OAM–Airy beams: Stacking cubic and $q=7$ fork gratings produces a superposed intensity map (Airy lobe fringes intersected by an $\ell=7$ OAM doughnut). Alignment of the fork grating relative to the main Airy lobe demonstrates spatial control over the hybrid envelope.

The experimental data exhibit near-quantitative agreement with Fresnel–Kirchhoff integral simulations for the output structure and intensity profiles. The estimated cross-talk between OAM channels is below 10%, shown by the spatial separation of annular (OAM) and lobe-like (Airy) structures.

Diffraction efficiency per OAM mode, defined as $\eta_\ell = I_\ell / I_\mathrm{inc}$, is limited to $\sim$1% per fork grating in single-pass; stacking allows use of otherwise "lost" beam intensity to populate additional modes without further loss in individual channels.

4. Applications in Quantum Sensing and Scattering

Neutron Airy and OAM–Airy beams admit multiple uses:

• Neutron OAM spectroscopy: Measurement of the full $\ell$-distribution pre- and post-sample enables direct paper of angular momentum selection rules and entanglement transfer in magnetic, chiral, or topological materials. This is analogous to OAM-resolved EELS in electron microscopy.
• Probing topological and auto-focusing beams: Helical ($\ell\ne0$) beams selectively couple to skyrmions and other mesoscale spin textures, while Airy beams, due to auto-focusing properties, can focus at variable depths within a sample—probing dynamics or gradients inaccessible to conventional plane-wave neutrons.
• 2D multiplexing: Combined use of gratings and pulsed neutron time-of-flight demultiplexing allows for simultaneous energy–$\ell$ resolved measurements, providing energy- and OAM-resolved scattering data in a single acquisition.
• Time-resolved and differential studies: Orthogonal arrangement of $q=0$ and $q\neq0$ gratings on the same detector supports parallelized control and readout schemes, relevant for studies of irreversible phase transitions or polarization-dependent processes.

These capabilities supplement and dramatically extend conventional small-angle neutron scattering (SANS) methods.

5. Relation to Spin-Textured and Multimode Entangled Beams

Neutron Airy beams, while not intrinsically spin-textured, are synergistically implemented with spin–OAM entangling techniques described in the context of magnetic Wollaston prisms (MWPs) (Thien et al., 2022). In particular:

• Integration of MWPs and multi-grating stacks enables the engineering of combined spatial (Airy), OAM, and spin-textured beams.
• The emerging multimode operator formalism treats OAM or Airy indices as additional "qubits" within the tensor-product space, facilitating design and analysis of structured entangled neutron interferometry (Lu et al., 2019).

A plausible implication is the possibility to entangle spin, path, energy, OAM, and Airy modes, realized through a combination of MWPs, RF spin flippers, and phase-grating elements, for use in multiparameter quantum sensing and contextuality violation experiments.

6. Performance Metrics, Trade-offs, and Practical Considerations

Several performance aspects are noted (Lailey et al., 10 Nov 2025):

Parameter	Typical Value / Consideration	Effect
Grating period	$120$ nm	Sets mode spacing and diffraction efficiency
Diffraction $\eta$	$\sim 0.5$–$1$% per mode per grating	Stacking recovers efficiency in multiple modes
Cross-talk	$<10\%$ between OAM/Airy channels	Allows unambiguous channel deconvolution
Coherence length	$\sim10$ $\mu$m (transverse)	Must exceed kinoform feature size
Detector	$1$ m$^2$, pixel $\lesssim 6$ mm	Resolves rings/lobes up to $\ell=7$

Trade-offs include the reduced efficiency per channel for deep OAM (high $q$) or strongly focused Airy states; the stacking solution is favorable compared to extremely deep single-phase structures. The technique benefits from high transverse coherence and precise sample–grating alignment.

7. Outlook and Future Directions

The structured neutron Airy/OAM–Airy modality opens multiple directions:

• Design of dynamically switchable multimode sources for quantum neutron optics.
• Probing nonlinear and topological scattering phenomena, including studies of skyrmion dynamics or Weyl semimetal Fermi arcs.
• Incorporation into generalized multimode entangled neutron interferometry platforms, potentially including contextuality witness and van Hove–type quantum linear response protocols.
• Further exploration of Airy–OAM mode-coupling and auto-focusing effects with variable sample environments.

In summary, neutron Airy beams and their hybridizations with OAM states constitute a new class of quantum probes, enabling advanced spatial and angular momentum control in neutron scattering, materials spectroscopy, and quantum information applications (Thien et al., 2022, Lailey et al., 10 Nov 2025, Lu et al., 2019).