---
title: Fractional Orbital Angular Momentum
url: https://www.emergentmind.com/topics/fractional-orbital-angular-momentum
type: topic
---

# Fractional Orbital 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 [1605.03899] [1106.1571] [2102.08879].

## 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 [1106.1571] [1810.05135].

When $\ell$ is non-integer, a full azimuthal circuit no longer yields an integer multiple of $2\pi$, 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 $\ell$; instead, they are superpositions of integer-OAM eigenmodes. In one explicit fractional-vortex decomposition, the modal weights are
$$
P_{m'}(M)=\frac{\sin^2(\pi M)}{\pi^2(M-m')^2},
$$
with mean OAM
$$
\bar M=M-\frac{\sin(2\pi M)}{2\pi},
$$
showing directly that the nominal fractional charge $M$ and the mean OAM need not coincide [1605.03899] [1804.09437].

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
$$
\psi(2\pi)=e^{i\theta}\psi(0),\qquad \delta=\frac{\theta}{2\pi},
$$
which gives angular eigenvalues
$$
M=\delta+m,\qquad m\in\mathbb Z.
$$
For the Pauli phase $\theta=\pi$, 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 [2102.08879].

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 $\alpha(\rho,\phi)$, a circularly polarized input of handedness $\sigma=\pm1$ acquires
$$
\Delta\Phi(\rho,\phi)=2\sigma\alpha(\rho,\phi),
$$
and the cross-polarized output obeys
$$
E_{\text{out}}^{(-\sigma)}(\rho,\phi)\propto T_{\text{cross}}e^{i2\sigma\alpha(\rho,\phi)}E_{\text{in}}^{(\sigma)}(\rho,\phi).
$$
The topological charge follows from
$$
\ell=\frac{d\Delta\Phi}{d\phi}=2\sigma\frac{d\alpha}{d\phi}.
$$
For the canonical profile $\alpha(\phi)=q\phi+\alpha_0$, one obtains $\ell=2\sigma q$, so fractional $q$ directly yields fractional $\ell$ [1605.03899].

Dielectric metasurfaces based on TiO$_2$ nanofins were used to implement this mechanism in the visible, with nominal nanofin dimensions of length $\approx 250$ nm, width $\approx 90$ nm, height $\approx 600$ nm, radial spacing $\approx 325$ nm, and device diameters typically $500\,\mu$m. At 532 nm, these devices acted as local half-wave retarders and reached absolute conversion efficiency up to $\approx 60\%$. The same platform generated integer vortices with $\ell=1,2,5,10$ and a fractional beam with nominal $\ell=6.5$, and also enabled simultaneous collinear generation of $\ell=5$ and $\ell=10$ by interleaving two $q$-profiles in alternating radial rows [1605.03899].

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 $l=-3$, fractional $l=-1.5$, and high-order $l=-10$ 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 $+y$ [1911.01033].

Fractional-vortex beam families generated by rotationally symmetric superposition provide a different construction. For the $n$-fold superposition
$$
|M_n'\rangle=\sum_{k=0}^{n-1}\hat U\!\left(\frac{2\pi k}{n}\right)|M\rangle,
$$
the mean OAM becomes
$$
\bar M_n(M)=M-\frac{n}{2\pi}\sin\!\left(\frac{2\pi M}{n}\right),
$$
and the associated phase-dislocation metric is
$$
\frac{d\bar M_n}{dM}=1-\cos\!\left(\frac{2\pi M}{n}\right).
$$
This construction was used to formulate a quantifiable complementarity between OAM noneigenvalue and angular-position variation [1804.09437].

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 [1708.01784].

## 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 [1605.03899].

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
$$
x_{\text{cms}}=\sum_i x_i\cdot I(x_i),\qquad
V=\sum_i (x_i-x_{\text{cms}})^2\cdot I(x_i)
$$
are used as quantitative proxies. In that implementation, $x_{\text{cms}}$ is proportional to the average OAM, while the sorted variance $V$ measures spectral width or peak splitting. Integer states remain rotation-invariant in the sorter, whereas fractional states depend strongly on the branch-cut orientation $\theta$: at $\theta=0^\circ$ they appear as single spots between adjacent integers, and at $\theta=180^\circ$ they produce symmetric double peaks and strongly increased $V$. The same protocol distinguishes intrinsic OAM from total OAM, because in that geometry $\theta=0^\circ$ measures intrinsic OAM and $\theta=180^\circ$ measures total OAM [1810.05135].

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 $\ell=1/2$ per photon, but transverse displacement of the phase plate moved the singularity off the beam axis and reduced the expectation value continuously according to
$$
\langle L_z\rangle/\hbar \approx Q\,\exp\!\left(-\frac{2r_0^2}{w_0^2}\right),
$$
with $Q=1/2$ in the experiment. In a Laguerre–Gaussian expansion,
$$
\langle L_z\rangle/\hbar=\sum_{l,p}l\,|C_{lp}|^2.
$$
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 [1106.1571].

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 [1810.05135] [1106.1571].

## 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,
$$
P^{(2)}(2\omega)\propto \chi^{(2)}E_1(\omega)E_2(\omega),
$$
so the OAM addition rule is
$$
\ell_{\text{SH}}=\ell_1+\ell_2.
$$
In the specific configuration studied, the two input beams carried equal-magnitude and opposite-sign fractional OAM, $\ell_1=+\ell(r_0)$ and $\ell_2=-\ell(r_0)$, 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 [1106.1571].

Propagation itself can smooth or redistribute fractional signatures. In microwave reflection metasurfaces, the near-field fractional $l=-1.5$ 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 [1911.01033].

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 $2a$, and a protected superdomain wall introduces the required phase discontinuity. The diffracted OAM index follows
$$
\ell=\frac{\mathbf Q\cdot \mathbf t}{2\pi}.
$$
At structural Bragg peaks this gives
$$
\ell_{\text{struct}}=qK\in\mathbb Z,
$$
whereas at antiferromagnetic magnetic peaks it gives
$$
\ell_{\text{mag}}=\frac{qK'}{2},
$$
which is half-integer for odd $q$. Experiments at the Fe $L_3$ edge observed integer OAM at charge peaks and fractional OAM at magnetic peaks, with fitted effective indices $a\approx 0.576\pm0.002$ at $(H,K)=(1/2,-1/2)$ and $a\approx 1.569\pm0.013$ at $(3/2,-1/2)$. At 340 K, the branch-cut angle $\phi_0$ fluctuated by up to 1.34 radians within a single 0.3 s frame, corresponding to real-time rotation of the fractional OAM beam [2606.03625].

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
$$
\psi(\phi+2\pi)=e^{i\gamma}\psi(\phi),
$$
the orbital quantum number becomes
$$
m=n+\frac{\gamma}{2\pi}.
$$
For the Pauli phase $\gamma=\pi$, one has $m\in\mathbb Z+\tfrac12$. 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 $m=p\pi/\Theta$, while in knife-edge diffraction the expansion contains Bessel functions of order $n/2$, mixing integer and half-integer angular content. In circular quantum dots, the Pauli principle selects integer orbital quantization for even $N$ and half-integer quantization for odd $N$ [2102.08879].

Electron vortex beams furnish a geometric-phase realization. In the skyrmionic model used for electron vortex beams, the Berry phase is $\gamma=2\pi\mu$, and for a vortex line tilted by angle $\theta$ the effective monopole charge is
$$
\mu(\theta)=\frac12(1-\cos\theta).
$$
The orbital and spin expectations are shifted by spin–orbit interaction according to
$$
\langle \vec L\rangle=(\ell-\mu)\,\hat{\mathbf z},\qquad
\langle \vec S\rangle=(s+\mu)\,\hat{\mathbf z}.
$$
Because $\mu$ 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 $d\mu/d\ln L\le 0$, whereas a longitudinal magnetic field modifies the effective monopole charge to
$$
\mu_{\mathrm{eff}}=\mu+g(2p+|\ell|+1),
$$
allowing stabilization through Landau-type Laguerre–Gaussian structure and Gouy-phase engineering [1608.03562].

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
$$
f_S(\beta)=\frac{1+3\beta^2}{3+\beta^2},\qquad
f_L(\beta)=\frac{2-2\beta^2}{3+\beta^2},
$$
with $s=\pm \tfrac12$. 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 $\langle L_3\rangle=0$. 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 [2508.17395].

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
$$
L_z=\hbar\left(n+\frac12\right)+\frac{\mu\lambda}{2\pi\varepsilon_0 c^2}.
$$
An alternative Aharonov–Casher-only construction gives mechanical angular momentum
$$
J_{K,n}=n\hbar-\frac{\mu\lambda}{2\pi\varepsilon_0 c^2}.
$$
The distinction between canonical and mechanical angular momentum is therefore substantive, not merely terminological [1805.09854].

A more formal proposal based on discretized SO(3) modifies the angular-momentum generator itself. With minimum measurable angle $\Delta\phi$, the paper defines
$$
\beta=\frac{\sin(\Delta\phi)}{\Delta\phi},
$$
and obtains magnetic quantum numbers
$$
m_f=n\hbar\beta.
$$
In that scheme, orbital angular momentum becomes fractional because the generator of rotations is altered by the assumed angle discretization [1504.01673].

## 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
$$
l_{\text{eff}}=M_{\text{eff}}=M-\nu,
$$
the effective magnetic field is
$$
B^*=(1-\nu)B,
$$
and the corresponding magnetic length is
$$
\ell^{*2}=\frac{\ell_B^2}{1-\nu}.
$$
Inside the droplet, the quantized orbit radii obey
$$
R^2=2(M-\nu)\ell^{*2},
$$
while the short-distance pair correlation behaves as
$$
g(r)\sim \left(\frac{r}{\ell^*}\right)^{2(M-\nu)}.
$$
The exchange phase is $e^{i\pi\nu}$, so the fractional part of relative angular momentum directly encodes braid statistics [1409.6251].

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 $\pi$ 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 [1605.03899] [1810.05135] [2606.03625] [1708.01784].

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 [1106.1571] [2102.08879] [1805.09854] [2508.17395].

Source: https://www.emergentmind.com/topics/fractional-orbital-angular-momentum