Helical Neutron Wavefronts: OAM in Neutron Science

• Helical neutron wavefronts are defined by an azimuthally winding phase factor that imparts quantized orbital angular momentum and creates doughnut-shaped far-field profiles.
• They are experimentally generated using fork-dislocation phase gratings that ensure coherence matching and enable interferometric recovery of phase information in SANS.
• Their unique structure allows decoupling the transverse coherence length from the single-particle wavepacket size, leading to novel state-dependent neutron scattering observations.

Helical neutron wavefronts are neutron matter-wave states whose phase winds azimuthally around the propagation axis, typically in the form $\psi(r,\phi,z)\propto e^{i\ell\phi}$, so that each neutron carries orbital angular momentum (OAM) $\ell\hbar$ along the beam axis. The phase singularity on axis suppresses central intensity and gives rise, in the far field, to annular or doughnut-shaped diffraction profiles. Within neutron science, these states are also termed twisted neutrons or neutron helical waves. Recent work has established their experimental realization with fork-dislocation phase gratings, their interferometric readout in small-angle neutron scattering (SANS), and their use as probes of both neutron-state structure and neutron–matter interactions (Sarenac et al., 2022, Sarenac et al., 2024, Sarenac et al., 21 Jun 2026).

1. Definition and mathematical structure

In the neutron OAM literature, a helical neutron wavefront is the neutron analogue of an optical vortex beam: a wave packet whose transverse phase advances by $2\pi \ell$ under one circuit around the beam axis. A standard form is

$\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$

where $\ell$ is the topological charge or OAM quantum number and $\phi$ is the azimuthal angle. Such states carry quantized OAM $\ell\hbar$ per neutron and exhibit a phase singularity at the axis; because the wavefunction vanishes there, the far-field intensity is typically doughnut-shaped rather than centrally peaked (Sarenac et al., 2024).

Several complementary representations are used. In scattering theory, twisted neutrons are written as coherent superpositions of plane waves whose transverse momenta lie on a cone. In the Bessel-beam formulation,

$$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$

with plane-wave weights

$$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$

All plane-wave components therefore share the same $|\mathbf p_\perp|=\hbar\varkappa$, while their azimuthal angles vary; geometrically, their momenta lie on a cone of opening angle

$\ell\hbar$0

The factor $\ell\hbar$1 is the momentum-space signature of the helical phase winding (Afanasev et al., 2019).

In experimentally generated beams, the OAM state need not be a pure single-radial-mode Laguerre–Gaussian (LG) state. One experimentally relevant description is

$\ell\hbar$2

so the helical phase factor is sharp while the radial structure may be a superposition over $\ell\hbar$3 (Sarenac et al., 21 Jun 2026). This distinction matters experimentally because several observables, especially ring radius and radial broadening, depend on the radial content as well as on $\ell\hbar$4.

The introduction of OAM adds a neutron degree of freedom beyond spin and linear momentum. This has motivated both fundamental studies of state structure and applied studies in SANS, interferometry, and neutron scattering (Sarenac et al., 2022).

2. Preparation and first experimental realization

The experimentally established route to helical neutron wavefronts uses fork-dislocation phase gratings. These are holographic phase elements with an azimuthal dislocation that imprints a vortex phase on transmitted diffraction orders. One form of the grating profile is

$\ell\hbar$5

where $\ell\hbar$6 is the grating period and $\ell\hbar$7 is the fork topological charge. In the earlier experimental realization, the grating action is summarized as

$\ell\hbar$8

The $\ell\hbar$9-th diffraction order therefore carries the azimuthal phase factor $2\pi \ell$0 and OAM

$2\pi \ell$1

For the first diffraction order, $2\pi \ell$2 (Sarenac et al., 2022).

The first direct realization of neutron helical waves used a large $2\pi \ell$3D array of microfabricated fork-dislocation phase gratings rather than a single microscopic element. The reported array area was $2\pi \ell$4, containing $2\pi \ell$5 gratings, each of size $2\pi \ell$6, with grating period $2\pi \ell$7 nm. This array geometry was required because a single grating would yield too few neutron counts to measure efficiently (Sarenac et al., 2022).

A critical enabling condition was coherence matching. On the GP-SANS beamline at the High Flux Isotope Reactor, the source aperture diameter was $2\pi \ell$8 mm, the aperture-to-sample distance was $2\pi \ell$9 m, and the wavelength was $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$0. The transverse coherence length was estimated as

$\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$1

which is approximately the size of one grating element. The experiment was therefore designed so that the beam was coherent across each individual fork grating, preserving the phase structure required to generate neutron OAM (Sarenac et al., 2022).

Experimental validation relied on far-field SANS detection. The measurements, performed for grating topologies $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$2 and $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$3, showed visible $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$4 diffraction orders, and the first orders displayed the expected doughnut-shaped intensity profiles. Two reported signatures were central. First, the first-order diffraction angle obeyed

$\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$5

which for $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$6 Å and $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$7 nm gave $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$8 rad. Second, the azimuthally integrated intensity of the dominant $\psi(r,\phi,z)\propto A(r,z)e^{i\ell\phi},$9 mode peaked at

$\ell$0

so the ring radius increased with $\ell$1. The observed diffraction locations and doughnut-size scaling were consistent with these predictions, supporting the interpretation that the outgoing neutron beams carried helical wavefronts with well-defined OAM content (Sarenac et al., 2022).

3. Interferometric recovery of helical phase information in SANS

A major limitation of conventional SANS is that it records only far-field intensity and therefore loses phase information. In standard form,

$\ell$2

so the observed signal is the modulus squared of the Fourier transform of the outgoing wave. Whether the wavefront is flat, helical, or otherwise structured is therefore not directly accessible from intensity alone (Sarenac et al., 2024).

The interferometric advance reported in 2024 addresses this limitation by coherently superposing an array of reference beams with an array of object beams. The reference beams are engineered to have complementary phase profiles to the object beams, allowing strong interference in the far field despite the large beam size and limited transverse coherence typical of SANS. For helical neutron waves, the complementarity is realized by preparing opposite OAM states, $\ell$3 and $\ell$4, in corresponding diffraction orders (Sarenac et al., 2024).

The far-field interference then exhibits the characteristic petal structure of opposite-charge vortex superpositions. The azimuthal dependence is

$\ell$5

so the pattern contains $\ell$6 azimuthal maxima, and a relative phase shift $\ell$7 rotates the petals. The phase of the neutron helical wavefront thus becomes directly observable through a measurable intensity rotation (Sarenac et al., 2024).

The interferometric geometry imposes two practical conditions. First, object and reference beams must overlap collinearly at the detector. This is achieved by a translational offset between the two grating arrays,

$\ell$8

where $\ell$9 is the separation between arrays. If $\phi$0 exceeds the array period $\phi$1, it is taken modulo $\phi$2, and maximum contrast occurs when $\phi$3. Second, the transverse coherence length must exceed $\phi$4, because the interference is between first diffraction orders of the same incoming beam (Sarenac et al., 2024).

Experimentally, the method used a double-sided silicon wafer with fork-dislocation phase-grating arrays on both faces: one side with $\phi$5 and the other with $\phi$6. Each array covered $\phi$7 and contained millions of $\phi$8 grating elements, with period $\phi$9 nm, height $\ell\hbar$0 nm, and $\ell\hbar$1 spacing. The wafer thickness was $\ell\hbar$2. At the GP-SANS beamline at ORNL, with central wavelength $\ell\hbar$3 Å and $\ell\hbar$4, the overlap condition gave an estimated optimal translation of $\ell\hbar$5, corresponding to a rotation of roughly $\ell\hbar$6 mrad. By varying pitch and yaw of the wafer and recording patterns for $\ell\hbar$7 minutes per setting, the experiment produced the first observation of petal-structure signatures of helical neutron wave interference and, in the paper’s formulation, the first direct recovery of phase information from small-angle scattering using a neutron interferometric approach (Sarenac et al., 2024).

This established that neutron OAM states are not merely preparable but also phase-readable within a SANS-compatible geometry. A plausible implication is that structured neutron phase information can become a routine contrast channel in SANS rather than remaining hidden behind intensity-only observables.

4. Coherence length, divergence, and single-particle wavepacket extent

Helical neutron wavefronts have also been used to resolve a longstanding ambiguity in neutron optics: the distinction between transverse coherence length and the spatial extent of an individual neutron wavepacket. The 2026 study defines the transverse coherence length as an ensemble property governed by the angular spread of the beam, with representative scalings

$\ell\hbar$8

and, for a Gaussian angular distribution,

$\ell\hbar$9

By contrast, the wavepacket extent $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$0 is the transverse width of the single-neutron wavefunction $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$1 (Sarenac et al., 21 Jun 2026).

The central observation is that helical neutron states provide different sensitivities to these two quantities. The OAM phase suppresses on-axis intensity for $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$2, creating an annular profile. In the reported geometry, the radius of the annulus depends primarily on the single-particle extent $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$3, whereas the coherence length $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$4 mainly broadens the ring. This permits a separation that conventional Gaussian-like beam profiles do not provide (Sarenac et al., 21 Jun 2026).

The experiment was performed on SANS2D at ISIS with source aperture $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$5, collimation length $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$6, sample aperture $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$7, and sample-to-detector distance $$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$8. The geometric beam divergence was

$$\psi_{\varkappa m p_z \lambda}(\mathbf r) = \int \frac{d^2\mathbf p_\perp}{(2\pi)^2}\, a_{\varkappa m}(\mathbf p_\perp)\, w^{(\lambda)}(\mathbf p)\, e^{i\mathbf p\cdot \mathbf r/\hbar},$$9

The wavelength range was $$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$0–$$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$1, with the main data around $$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$2–$$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$3. Helical neutron states with

$$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$4

were generated and analyzed through radially integrated intensity profiles (Sarenac et al., 21 Jun 2026).

The reported scaling laws make the diagnostic role of OAM explicit. For a helically phased Gaussian-like neutron wavepacket,

$$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$5

and for a fork-dislocation phase-grating array geometry,

$$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$6

where $$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$7 is the grating size or characteristic width in that configuration. The paper further notes that, although a pure $$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$8 LG mode would exhibit a peak radius scaling roughly as $$a_{\varkappa m}(\mathbf p_\perp) = (-i)^m e^{im\varphi}\, \frac{2\pi}{p_\perp}\, \delta(p_\perp-\hbar\varkappa).$$9, the experimentally relevant radial-mode superposition leads to an approximately linear growth with $|\mathbf p_\perp|=\hbar\varkappa$0 (Sarenac et al., 21 Jun 2026).

The quantitative results were a beam divergence of about $|\mathbf p_\perp|=\hbar\varkappa$1 mrad, a transverse coherence length of about $|\mathbf p_\perp|=\hbar\varkappa$2 nm at $|\mathbf p_\perp|=\hbar\varkappa$3, and a lower bound

$|\mathbf p_\perp|=\hbar\varkappa$4

on the spatial extent of the individual neutron wavepackets. The inferred wavepacket extent was therefore more than an order of magnitude larger than the coherence length (Sarenac et al., 21 Jun 2026).

This directly contradicts the common identification of coherence length with single-particle size. In the paper’s interpretation, collimation and post-selection can increase the measured coherence length of the ensemble, but this does not imply that the underlying single-neutron wavepacket has become larger. Helical neutron wavefronts make that distinction experimentally visible because the ring radius tracks $|\mathbf p_\perp|=\hbar\varkappa$5 while the ring broadening tracks $|\mathbf p_\perp|=\hbar\varkappa$6 (Sarenac et al., 21 Jun 2026).

5. Scattering by nuclei and state-dependent observables

The use of helical neutron wavefronts in scattering theory predates their full experimental maturation and shows that neutron OAM modifies even standard low-energy processes. In the Schwinger problem, the plane-wave scattering amplitude for a neutron scattered by a nucleus is written as

$|\mathbf p_\perp|=\hbar\varkappa$7

with

$|\mathbf p_\perp|=\hbar\varkappa$8

For plane-wave incidence, the well-known result depends on transverse polarization and on the imaginary part of the nuclear amplitude, but not on longitudinal polarization or helicity (Afanasev et al., 2019).

For a single twisted neutron incident on a macroscopic target, the angular distribution is altered by the OAM structure. The scattering becomes an average over coherent plane-wave components on the momentum cone, and the singular behavior shifts from $|\mathbf p_\perp|=\hbar\varkappa$9 to

$\ell\hbar$00

where $\ell\hbar$01 is the cone angle of the twisted beam. The paper describes the resulting angular distribution as exhibiting a step-like change around $\ell\hbar$02 (Afanasev et al., 2019).

For a coherent superposition of two twisted states with different OAM values,

$\ell\hbar$03

the cross section acquires dependence on the OAM difference $\ell\hbar$04, the relative phase $\ell\hbar$05, and the longitudinal polarization component $\ell\hbar$06. This is absent in ordinary plane-wave Schwinger scattering. The paper gives the longitudinal polarization asymmetry

$\ell\hbar$07

The reported estimate for thermal neutrons and gold places this asymmetry at the level of a few ppm (Afanasev et al., 2019).

The most distinctive effects arise for localized targets. For a single nucleus at impact parameter $\ell\hbar$08 relative to the beam axis, the twisted-neutron scattering amplitude becomes position dependent,

$\ell\hbar$09

After angular averaging, the distribution takes the form

$\ell\hbar$10

with helicity asymmetry

$\ell\hbar$11

In this localized-target regime, the observables depend on neutron helicity and on the real part $\ell\hbar$12 of the nuclear amplitude, thereby giving access to the phase $\ell\hbar$13 already in the Born approximation. The paper estimates $\ell\hbar$14 for realistic parameter ranges and summarizes the overall predicted spin asymmetries as ranging from about $\ell\hbar$15 up to $\ell\hbar$16 (Afanasev et al., 2019).

These results imply that helical neutron wavefronts are not only a beam-shaping novelty. They alter the information content of neutron scattering itself, making the outcome sensitive to OAM content, superposition phase, longitudinal polarization, helicity, and target localization. In the paper’s terms, this opens routes to quantum tomography of neutron states as well as applications in hadronic studies, low-energy nuclear physics, tests of fundamental symmetries, and neutron optics (Afanasev et al., 2019).

6. Relation to broader helical-wave physics and terminological boundaries

Helical neutron wavefronts are part of a broader structured-wave landscape, but several neighboring concepts should be distinguished carefully. In optics, helical wavefronts are carried by OAM beams, often represented by Laguerre–Gaussian modes with characteristic phase factor $\ell\hbar$17. Optical work on helical dichroism defines the differential response to opposite topological charges and emphasizes that $\ell\hbar$18 and $\ell\hbar$19 have opposite helical handedness while retaining equal $\ell\hbar$20 (Pan et al., 22 Jan 2025). This provides a close analogue for the neutron case at the level of phase topology, although the neutron papers considered here focus on SANS phase recovery, state preparation, coherence diagnostics, and scattering rather than on a neutron-specific helical dichroism observable.

A related symmetry-based line of work in X-ray theory constructs twisted waves as eigenfunctions of the continuous helical group and shows that, when scattered from helical structures, they obey a twisted analogue of the Von Laue condition and yield discrete diffraction patterns. This suggests that symmetry-matched helical illumination can reorganize reciprocal-space selection rules when the incoming probe and the target share helical structure (Friesecke et al., 2015). A plausible implication is that increasingly controlled neutron OAM beams may support analogous symmetry-tailored neutron scattering strategies, although the X-ray formulas themselves are electromagnetic rather than neutron specific.

Helical structure also appears in electromagnetic near fields. In helical evanescent waves, a helicity-dependent momentum-space geometry ties together field orientation, spin, and Poynting vector through helicity-dependent half tangent lines in momentum space, and zeros in the angular spectrum are interpreted as helicity singularities (Wei et al., 2019). This is conceptually adjacent to neutron OAM because both settings treat helicity or OAM as organizing variables for structured-wave propagation, but the underlying field theory and observables differ.

The terminology “helical” can also be misleading if imported from condensed-matter contexts. The predicted helical phase of confined chiral $\ell\hbar$21-wave superfluid $\ell\hbar$22He is a static equilibrium texture in which the order parameter rotates along the channel axis, retaining a combined rotation–translation symmetry. It is not a propagating neutron wavefront or a neutron-optical OAM state (Wiman et al., 2018). Distinguishing these usages avoids a common category error: helical neutron wavefronts are states of propagating neutron matter waves, whereas helical order in confined superfluids refers to symmetry-broken condensate textures.

Taken together, the neutron literature establishes helical neutron wavefronts as experimentally accessible OAM-carrying states with distinctive phase topology, annular far-field structure, and nontrivial consequences for interferometry and scattering. Their demonstrated roles in SANS phase recovery, separation of coherence from wavepacket extent, and state-sensitive nuclear scattering indicate that neutron OAM is becoming a practical degree of freedom in neutron science rather than a formal extension of optical vortex concepts alone.