Fractional Orbital Angular Momentum

• Fractional orbital angular momentum is defined by non-integer helical phase profiles that create branch cuts and yield superpositions of integer eigenstates.
• Structured-light techniques exploit the Pancharatnam–Berry phase to generate fractional OAM beams via space-variant optical elements and metasurface engineering.
• The concept extends to quantum and condensed-matter systems where altered boundary conditions and effective gauge fields produce shifted spectral eigenvalues and modified angular momentum.

Fractional orbital angular momentum denotes non-integer orbital angular-momentum content associated with azimuthal wavefront structure, but the term does not refer to a single universal mechanism. In structured-light optics it usually denotes beams whose azimuthal phase is nominally proportional to $e^{i\ell\phi}$ with non-integer $\ell$, which therefore contain a branch cut and are not eigenstates of the OAM operator with a single integer eigenvalue; in off-axis-vortex optics it can denote a non-integer expectation value defined relative to a chosen axis; and in two-dimensional quantum or topological settings it can denote genuinely shifted orbital-angular-momentum spectra enforced by boundary conditions, Berry phases, or effective gauge fields (Devlin et al., 2016, Bovino et al., 2011, Krylov et al., 2021).

1. Mathematical definitions and competing notions of fractionality

For paraxial optical and matter-wave fields, the canonical starting point is the helical phase factor $e^{i\ell\phi}$. For integer $\ell$, the field has a phase winding of $2\pi \ell$ around the axis, and the $z$-component of orbital angular momentum is $L_z=\ell\hbar$ per photon or per particle in the corresponding eigenmode description. In optics, the operator form $L_z=-i\hbar\,\partial/\partial\phi$ is standard, and the topological charge can also be written as

$$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$

with $f$ the optical phase (Bovino et al., 2011, Berger et al., 2018).

When $\ell$0 is non-integer, a full azimuthal circuit no longer yields an integer multiple of $\ell$1, so the field must contain a discontinuity line or branch cut. This breaks rotational symmetry and introduces a preferred transverse direction. Such beams are not eigenstates of the OAM operator with a single integer $\ell$2; instead, they are superpositions of integer-OAM eigenmodes. In one explicit fractional-vortex decomposition, the modal weights are

$\ell$3

with mean OAM

$\ell$4

showing directly that the nominal fractional charge $\ell$5 and the mean OAM need not coincide (Devlin et al., 2016, Huang, 2018).

A separate, more topological notion appears in two-dimensional quantum mechanics. There, the angular generator can be defined with a self-adjoint extension specified by

$\ell$6

which gives angular eigenvalues

$\ell$7

For the Pauli phase $\ell$8, one obtains half-integer orbital quantization. In that usage, fractional orbital angular momentum is not a branch-cut beam property but a consequence of monodromy on a multiply connected configuration space (Krylov et al., 2021).

These distinctions are essential because the same phrase, “fractional orbital angular momentum,” may describe a non-eigenstate superposition, an axis-dependent expectation value, or a shifted operator spectrum. A plausible implication is that comparisons across optical, condensed-matter, and relativistic-electron literatures require explicit identification of which of these notions is being used.

2. Structured-light realizations and phase-engineering mechanisms

A major optical route to fractional OAM uses Pancharatnam–Berry geometric phase in spin-to-orbit converters. For a space-variant half-wave element with local optic-axis orientation $\ell$9, a circularly polarized input of handedness $e^{i\ell\phi}$0 acquires

$e^{i\ell\phi}$1

and the cross-polarized output obeys

$e^{i\ell\phi}$2

The topological charge follows from

$e^{i\ell\phi}$3

For the canonical profile $e^{i\ell\phi}$4, one obtains $e^{i\ell\phi}$5, so fractional $e^{i\ell\phi}$6 directly yields fractional $e^{i\ell\phi}$7 (Devlin et al., 2016).

Dielectric metasurfaces based on TiO$e^{i\ell\phi}$8 nanofins were used to implement this mechanism in the visible, with nominal nanofin dimensions of length $e^{i\ell\phi}$9 nm, width $\ell$0 nm, height $\ell$1 nm, radial spacing $\ell$2 nm, and device diameters typically $\ell$3m. At 532 nm, these devices acted as local half-wave retarders and reached absolute conversion efficiency up to $\ell$4. The same platform generated integer vortices with $\ell$5 and a fractional beam with nominal $\ell$6, and also enabled simultaneous collinear generation of $\ell$7 and $\ell$8 by interleaving two $\ell$9-profiles in alternating radial rows (Devlin et al., 2016).

Wideband reflection-type metasurfaces extend the same general phase-engineering logic to microwaves. A single-layer design based on deformed square-loop meta-atoms exploited PB phase in reflection and generated integer $2\pi \ell$0, fractional $2\pi \ell$1, and high-order $2\pi \ell$2 over a broad band. The reported operation covered 6.75 to 21.85 GHz for one meta-atom design, and the fractional case exhibited the expected branch-cut asymmetry as a notched doughnut with amplitude zeros preferentially along $2\pi \ell$3 (Yang et al., 2019).

Fractional-vortex beam families generated by rotationally symmetric superposition provide a different construction. For the $2\pi \ell$4-fold superposition

$2\pi \ell$5

the mean OAM becomes

$2\pi \ell$6

and the associated phase-dislocation metric is

$2\pi \ell$7

This construction was used to formulate a quantifiable complementarity between OAM noneigenvalue and angular-position variation (Huang, 2018).

Spiroid vortex beams introduce yet another beam family. In fractional-order hypergeometric-Gaussian spiroid beams, the OAM as a function of fractional-order topological charge forms chains of super-pulses, including sharp bursts or dips, with the pulse shape controlled by spiral torsion parameters in the angular spectrum. The reported phenomenon was explicitly proposed for optical switches and triggers (Volyar et al., 2017).

3. Diagnostics, modal analysis, and the role of axis choice

Interference remains the classical diagnostic for fractional optical singularities. Tilted-beam interferometry with a Gaussian reference yields pitchfork fringes; for fractional beams, a singularity line appears along which alternating single-charge fork dislocations are observed. Collinear interference with a Gaussian produces a spiral whose number of arms reflects the nearest integer charge, while the spiral orientation flips with input handedness (Devlin et al., 2016).

OAM sorting provides a more quantitative route. In a sorter based on the Berkhout method, the recorded two-dimensional intensity is integrated along the non-sorting axis, and the moments

$2\pi \ell$8

are used as quantitative proxies. In that implementation, $2\pi \ell$9 is proportional to the average OAM, while the sorted variance $z$0 measures spectral width or peak splitting. Integer states remain rotation-invariant in the sorter, whereas fractional states depend strongly on the branch-cut orientation $z$1: at $z$2 they appear as single spots between adjacent integers, and at $z$3 they produce symmetric double peaks and strongly increased $z$4. The same protocol distinguishes intrinsic OAM from total OAM, because in that geometry $z$5 measures intrinsic OAM and $z$6 measures total OAM (Berger et al., 2018).

Off-axis-vortex beams expose a different measurement subtlety. In noncollinear second-harmonic-generation studies, a half-integer spiral phase plate centered on a Gaussian beam produced nominal $z$7 per photon, but transverse displacement of the phase plate moved the singularity off the beam axis and reduced the expectation value continuously according to

$z$8

with $z$9 in the experiment. In a Laguerre–Gaussian expansion,

$L_z=\ell\hbar$0

Here the fractional value arises from modal mixing about the chosen axis, not from a single non-integer local vortex charge. The distinction between intrinsic and extrinsic OAM is therefore operationally crucial (Bovino et al., 2011).

A recurrent misconception is that any non-integer value read out by an instrument must indicate a “fractional vortex” in the branch-cut sense. The combined evidence from sorter-based measurements and off-axis-vortex studies indicates otherwise: non-integer values may arise either from a genuine branch-cut state or from evaluating an otherwise ordinary singularity about an axis displaced from the beam’s center of mass (Berger et al., 2018, Bovino et al., 2011).

4. Nonlinear conversion, propagation effects, and X-ray fractional OAM

Fractional OAM survives nonlinear optics, but not always in the most naive way. In noncollinear type-I second-harmonic generation with two specular off-axis vortex pumps, the second-order polarization multiplies the two fundamental fields,

$L_z=\ell\hbar$1

so the OAM addition rule is

$L_z=\ell\hbar$2

In the specific configuration studied, the two input beams carried equal-magnitude and opposite-sign fractional OAM, $L_z=\ell\hbar$3 and $L_z=\ell\hbar$4, so the generated second harmonic always had zero OAM even though the near-field and far-field morphologies evolved strongly with displacement. The spatial patterns developed nodal lines and multi-lobed structures, but the transverse Poynting vector exhibited no rotation, consistent with vanishing screw-type OAM (Bovino et al., 2011).

Propagation itself can smooth or redistribute fractional signatures. In microwave reflection metasurfaces, the near-field fractional $L_z=\ell\hbar$5 mode showed asymmetric, notched intensity and branch-cut phase structure, whereas in the far field the notch became less conspicuous and the OAM content spread into neighboring fractional orders. The same study explicitly reported lower far-field purity for fractional modes than for integer modes, making fractional states less suitable for long-distance transmission in that implementation (Yang et al., 2019).

Soft X-ray fractional OAM has now been demonstrated through coherent magnetic scattering from artificial spin ice. In square-lattice ASI with odd-charge edge dislocations, antiferromagnetic order doubles the magnetic period to $L_z=\ell\hbar$6, and a protected superdomain wall introduces the required phase discontinuity. The diffracted OAM index follows

$L_z=\ell\hbar$7

At structural Bragg peaks this gives

$L_z=\ell\hbar$8

whereas at antiferromagnetic magnetic peaks it gives

$L_z=\ell\hbar$9

which is half-integer for odd $L_z=-i\hbar\,\partial/\partial\phi$0. Experiments at the Fe $L_z=-i\hbar\,\partial/\partial\phi$1 edge observed integer OAM at charge peaks and fractional OAM at magnetic peaks, with fitted effective indices $L_z=-i\hbar\,\partial/\partial\phi$2 at $L_z=-i\hbar\,\partial/\partial\phi$3 and $L_z=-i\hbar\,\partial/\partial\phi$4 at $L_z=-i\hbar\,\partial/\partial\phi$5. At 340 K, the branch-cut angle $L_z=-i\hbar\,\partial/\partial\phi$6 fluctuated by up to 1.34 radians within a single 0.3 s frame, corresponding to real-time rotation of the fractional OAM beam (Montgomery et al., 2 Jun 2026).

This suggests that fractional OAM is not confined to visible-light beam shaping. It can also be encoded by defect topology and magnetic superstructure in reciprocal space, with domain-wall dynamics acting directly on the branch-cut orientation.

5. Two-dimensional quantum systems, electrons, and operator-level fractionality

In two-dimensional wave mechanics, fractional orbital angular momentum may be built into the operator domain rather than into a branch-cut field profile. With boundary condition

$L_z=-i\hbar\,\partial/\partial\phi$7

the orbital quantum number becomes

$L_z=-i\hbar\,\partial/\partial\phi$8

For the Pauli phase $L_z=-i\hbar\,\partial/\partial\phi$9, one has $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$0. This framework was applied to Helmholtz, Schrödinger, and Dirac problems, including wedge diffraction, knife-edge Fresnel diffraction, few-electron quantum dots, and graphene with overcharged impurities. In wedge geometries the angular quantization becomes $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$1, while in knife-edge diffraction the expansion contains Bessel functions of order $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$2, mixing integer and half-integer angular content. In circular quantum dots, the Pauli principle selects integer orbital quantization for even $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$3 and half-integer quantization for odd $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$4 (Krylov et al., 2021).

Electron vortex beams furnish a geometric-phase realization. In the skyrmionic model used for electron vortex beams, the Berry phase is $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$5, and for a vortex line tilted by angle $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$6 the effective monopole charge is

$$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$7

The orbital and spin expectations are shifted by spin–orbit interaction according to

$$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$8

Because $$Q=\frac{1}{2\pi}\oint \nabla f\cdot d\boldsymbol{\ell},$$9 is non-quantized for tilted vortices, the OAM becomes fractional. The same work argued that free-space fractional states are unstable under an RG flow $f$0, whereas a longitudinal magnetic field modifies the effective monopole charge to

$f$1

allowing stabilization through Landau-type Laguerre–Gaussian structure and Gouy-phase engineering (Bandyopadhyay et al., 2016).

Relativistic Dirac theory introduces a different sense of fractional orbital contribution. In a three-dimensional harmonic-oscillator model, one family of Dirac solutions contains a ground state for which the measured spin projection is partitioned between spin and orbital operators as

$f$2

with $f$3. At rest, two-thirds of the spin projection comes from OAM and one-third from the spin matrix; in the ultrarelativistic limit the OAM contribution vanishes. A second, equally energetic solution has $f$4. Here “fractional orbital angular momentum” means a fractional orbital contribution to what is measured as spin, rather than a non-integer OAM eigenvalue in the optical-vortex sense (Ducharme et al., 24 Aug 2025).

Cold-atom realizations supply operator spectra with explicit fractional shifts. For a cold neutral atom with permanent magnetic dipole moment in two electric fields and a harmonic trap, cooling to the lowest kinetic-energy level yields canonical angular-momentum eigenvalues

$f$5

An alternative Aharonov–Casher-only construction gives mechanical angular momentum

$f$6

The distinction between canonical and mechanical angular momentum is therefore substantive, not merely terminological (Jing et al., 2018).

A more formal proposal based on discretized SO(3) modifies the angular-momentum generator itself. With minimum measurable angle $f$7, the paper defines

$f$8

and obtains magnetic quantum numbers

$f$9

In that scheme, orbital angular momentum becomes fractional because the generator of rotations is altered by the assumed angle discretization (Abutaleb, 2015).

6. Relative angular momentum, applications, and unresolved distinctions

Fractionality also appears in two-particle relative motion. In a bosonic fractional quantum Hall droplet with two test bosons bound to vortices of the background fluid, the pair behaves as anyons. The effective relative angular momentum is shifted to

$\ell$00

the effective magnetic field is

$\ell$01

and the corresponding magnetic length is

$\ell$02

Inside the droplet, the quantized orbit radii obey

$\ell$03

while the short-distance pair correlation behaves as

$\ell$04

The exchange phase is $\ell$05, so the fractional part of relative angular momentum directly encodes braid statistics (Zhang et al., 2014).

Across applications, fractional OAM functions as an additional degree of freedom but also introduces sensitivity to geometry. Half-odd-integer optical beams with singularity lines rotated by $\ell$06 are orthogonal, a property already used in high-dimensional photon-entanglement experiments. Fractional optical beams and multi-OAM collinear generation have been linked to quantum communications, microscopy, vector-beam shaping, optical tweezers, and micromanipulation. OAM sorting with orientation-sensitive fractional-state readout was proposed for spectroscopy of exciton-polaritons in microcavities. In X-ray scattering, defect-encoded fractional OAM offers beam shaping, OAM multiplexing, and metrology of superdomain walls. Spiroid beams were proposed for optical switches and triggers because their OAM-versus-charge curves develop controllable super-pulses (Devlin et al., 2016, Berger et al., 2018, Montgomery et al., 2 Jun 2026, Volyar et al., 2017).

The principal unresolved issue is conceptual rather than merely technical. The literature supports at least four non-equivalent meanings of “fractional orbital angular momentum”: a branch-cut beam that is a superposition of integer OAM modes; an axis-dependent non-integer expectation value arising from displaced singularities; a genuine spectral shift produced by topological boundary conditions or effective gauge fields; and a fractional orbital contribution to another conserved quantity such as total spin. A plausible implication is that progress in the field depends less on a universal definition than on maintaining explicit distinctions among intrinsic and extrinsic OAM, canonical and mechanical angular momentum, single-particle and relative angular momentum, and eigenvalue spectra versus expectation values (Bovino et al., 2011, Krylov et al., 2021, Jing et al., 2018, Ducharme et al., 24 Aug 2025).