Anomalous Vortex Beams: Beyond Scalar Modes

• Anomalous vortex beams are wave fields that depart from conventional Laguerre–Gaussian modes by exhibiting features like fractional topological charge, non-uniform polarization, and spin–orbit non-separability.
• They are generated using advanced tools such as q-plates, metasurfaces, and holographic masks, which tailor phase structures to produce unique propagation phenomena including cusp caustics and space–time coupling.
• These beams enable applications in optical communications, high-resolution imaging, particle manipulation, and extend their impact to electron, acoustic, and matter-wave systems.

Anomalous vortex beams are vortex-carrying wave fields whose phase, polarization, propagation, caustic structure, or interaction dynamics depart substantially from the standard scalar Laguerre–Gaussian picture. In the optical case, the canonical reference is a field with azimuthal phase factor $e^{i l\phi}$, spatially uniform polarization, and orbital angular momentum (OAM) $l\hbar$ per photon; anomalous vortex beams instead include non-uniform polarization, non-separable spin–orbit states, cusp-caustic backgrounds, fractional and multi-singularity structures, space–time-coupled vortices, and interaction-induced topological rearrangements. The term also extends to electron, acoustic, atomic, and molecular matter-wave beams whenever vortex topology produces behavior absent in conventional scalar or plane-wave settings (Rodríguez-Fajardo et al., 2024).

1. Canonical reference and the meaning of anomaly

The standard monochromatic vortex beam is typically represented by a scalar envelope with azimuthal phase $e^{i l\phi}$, where $l$ is the topological charge and OAM index. In the usual paraxial optical setting, the field has a rotationally symmetric intensity distribution, often a ring, and a spatially homogeneous polarization. This structure underlies Laguerre–Gaussian and related textbook OAM modes, for which the phase winds by $2\pi l$ around the beam axis and the OAM per photon is $l\hbar$ (Yue et al., 2016).

Anomalous vortex beams are defined relative to that reference. In structured-light optics, they include vortex fields with non-uniform polarization, hybrid spin–orbit states, tailored topological charge combinations, nontrivial singularity structures, and propagation behavior not captured by cylindrically symmetric Laguerre–Gaussian evolution (Yue et al., 2016). In catastrophe-optics settings, a vortex embedded in a Pearcey–Gauss background rather than a Gaussian or Laguerre–Gaussian mode is anomalous because the underlying field exhibits cusp caustics, autofocusing, self-healing, and asymmetric scaling rather than simple self-similar ring expansion (Rodríguez-Fajardo et al., 2024). In wave scattering, the anomaly may lie not in the field profile itself but in the observables it unlocks: collisions of vortex beams allow direct access to phase differences of scattering amplitudes that are inaccessible in plane-wave experiments (Ivanov, 2012).

The same logic applies beyond optics. Electron vortex beams scattered by atomic columns exhibit reduced delocalization, site-symmetry-dependent splitting, and complex vortex-line fabrics rather than free-space straight vortex lines (Lubk et al., 2014). Acoustic vortex beams emitted by metamaterial apertures can display fractional or non-integer effective topological order, multi-singularity patterns, and frequency-tunable topology (Naify et al., 2016). Atomic and molecular vortex beams are anomalous in a different sense: the center-of-mass OAM belongs to composite particles and may couple to internal degrees of freedom such as molecular rotation or Rydberg excitation (Luski et al., 2021). This suggests that “anomalous” designates not a single beam family but a broad topological regime in which vortex-carrying fields cease to behave like standard scalar paraxial modes.

2. Topological charge, non-separability, and vortex constellations

The central invariant of a vortex field is the phase winding around a singularity. For a scalar field $\Psi(x,y)=|\Psi|e^{i\varphi(x,y)}$, the winding number is defined by

$$2\pi w \equiv \oint \nabla\varphi\cdot d\mathbf{s} = -i\oint \frac{\nabla\Psi}{\Psi}\cdot d\mathbf{s},$$

and the total winding number over a plane is the sum of all vortex charges (Lubk et al., 2014). This local topological quantity is distinct from OAM expectation values, which depend on the choice of axis and on system symmetry. In continuous rotational symmetry, the two are closely aligned; under discrete symmetry or in inhomogeneous media, OAM is generally not conserved even though total winding number is (Lubk et al., 2014).

A major class of anomalous beams arises when spatial and polarization degrees of freedom are non-separable. For q-plate-generated vector vortex beams, the Jones transformation couples spin angular momentum (SAM) and OAM so that a linearly polarized input becomes

$$\mathbf{E}(\mathbf{r}_\perp, 0) = \alpha\,\text{PeG}_{\ell=2q}(\mathbf{r}_\perp,0)\,e_L + \beta\,\text{PeG}_{\ell=-2q}(\mathbf{r}_\perp,0)\,e_R,$$

with left- and right-circular polarization components carrying opposite vortex charges (Rodríguez-Fajardo et al., 2024). Such fields cannot be factorized into a single spatial mode times a homogeneous polarization vector, and therefore belong to the broader class of classically entangled structured-light states (Rodríguez-Fajardo et al., 2024). A related construction appears in metasurface-generated vector vortex beams, where the output is a coherent superposition of a residual Gaussian component and a converted vortex component of orthogonal circular polarization, naturally represented on the hybrid-order Poincaré sphere (Yue et al., 2016).

Another anomalous topological regime is the vortex constellation. Reflection, scattering, or fabrication imperfections destabilize higher-order singularities, replacing a nominal charge-$\ell$ core by a cluster of unit-charge vortices. In a polynomial representation

$l\hbar$0

the roots $l\hbar$1 are the constellation coordinates. High-order aberrations then act not most naturally on the roots themselves but on the elementary symmetric polynomials of $l\hbar$2, which transform linearly under the aberration operator (Barros et al., 2023). This formulation makes explicit that anomalous vortex behavior often consists not in creating or destroying total topological charge but in redistributing it among singularities.

Fractional vortex beams constitute a further departure from the integer-$l\hbar$3 paradigm. Traditional fractional beams exhibit unit jumps in net topological charge when the source charge crosses a half-integer. The anomalous multi-ramp fractional vortex beam proposed in 2020 replaces the single-ramp phase profile by a multi-ramp spiral phase plate with independent control coefficients $l\hbar$4, yielding arbitrary topological-charge jumps at tunable critical thresholds (Zeng et al., 2020). This suggests that topological staircase behavior itself can be engineered.

3. Propagation anomalies: caustics, symmetry breaking, and space–time vortices

Propagation is one of the clearest diagnostics of anomalousity. Vortex Pearcey–Gauss beams are exemplary because the vortex is embedded in a cusp-caustic background rather than in a rotationally symmetric Gaussian. At the input plane,

$l\hbar$5

where $l\hbar$6 is the Pearcey integral describing a cusp caustic (Rodríguez-Fajardo et al., 2024). Experiments and simulations show asymmetric energy transfer across the cusp, far-field splitting of the characteristic Pearcey parabola into semi-parabolas or multiple branches, and splitting of $l\hbar$7 into two unit-charge vortices during propagation. These features are explicitly unlike the cylindrically symmetric ring expansion of Laguerre–Gaussian vortices (Rodríguez-Fajardo et al., 2024).

Discrete symmetry provides another route to anomalous propagation. When an electron vortex beam interacts with apertures of triangular or square symmetry, or with atomic columns of discrete site symmetry, the high-order vortex generally ceases to remain a single core. The cited symmetry criterion states that splitting becomes necessary when $l\hbar$8 for local $l\hbar$9-fold rotational symmetry [(Lubk et al., 2014); (Clark et al., 2016)]. The result is symmetry-constrained fragmentation into constellations of unit-charge vortices, often with additional vortex–antivortex pairs required to preserve total winding number. For example, an $e^{i l\phi}$0 electron vortex centered on a fourfold TiO column splits into four off-center $e^{i l\phi}$1 vortices, whereas an $e^{i l\phi}$2 beam on the same site can remain stable under perfect symmetry (Lubk et al., 2014).

The most radical propagation anomalies arise in space–time vortex beams. Starting from a monochromatic Bessel beam and applying either a transverse wave-vector shift or a Lorentz boost yields polychromatic solutions of the Klein–Gordon equation whose intrinsic OAM is not collinear with the mean momentum (Bliokh et al., 2012). In the shifted case, the spectrum remains a loop on the mass shell but becomes frequency dependent, producing time-diffracting spatio-temporal vortex beams; in the boosted case, the beam is non-diffracting in time but carries intrinsic OAM at an arbitrary angle to the propagation direction (Bliokh et al., 2012). This suggests that anomalous vortex structure can be interpreted as a four-dimensional generalization of the usual screw dislocation.

4. Generation platforms and engineered implementations

Anomalous vortex beams are generated by correspondingly nontrivial optical elements and wavefront-engineering schemes. Q-plates are central to spin–orbit conversion: a spatially patterned half-wave plate with topological charge $e^{i l\phi}$3 maps $e^{i l\phi}$4 and $e^{i l\phi}$5, thereby embedding opposite OAM components into orthogonal polarization channels (Rodríguez-Fajardo et al., 2024). In the Pearcey–Gauss realization, a spatial light modulator generates a scalar Pearcey–Gauss beam, and a q-plate converts it into either a scalar vortex Pearcey–Gauss beam or a vector vortex Pearcey–Gauss beam depending on the input polarization state (Rodríguez-Fajardo et al., 2024).

Metasurfaces provide a compact route to engineered anomalous states. The reflective plasmonic metasurface reported in 2016 uses spatially varying nanorod orientation

$e^{i l\phi}$6

to impose a Pancharatnam–Berry phase $e^{i l\phi}$7 on the converted circular-polarization component, while retaining a residual Gaussian component of opposite spin (Yue et al., 2016). By balancing converted and residual amplitudes, the device generates radially or azimuthally polarized vector vortex beams within a single subwavelength-scale element (Yue et al., 2016). This single-element generation strategy differs from earlier cascaded or interferometric approaches and makes the non-separable spin–orbit structure intrinsic to the device.

Compact generation can also proceed from incoherent sources. A holographic plasmonic nanostructure on a gold film, combined with spontaneous emission from PbS colloidal quantum dots, converts radially propagating surface plasmon polaritons into radially polarized vector vortex beams with non-zero OAM (Schanne et al., 2021). The anomaly here is the source mechanism: a structured vector vortex beam emerges from incoherent spontaneous emission after plasmon-mediated coherence transfer and holographic diffraction (Schanne et al., 2021).

Outside optics, anomalous vortex generation often exploits diffraction from singular masks. Atomic and molecular vortex beams were produced by diffracting supersonic helium atom and helium dimer beams from binary fork gratings fabricated on nanometric SiN membranes, yielding doughnut-shaped diffraction orders associated with discrete OAM states (Luski et al., 2021). Atomic Ferris wheel beams go further: a spiral light mask imprints an $e^{i l\phi}$8 phase and an $e^{i l\phi}$9 quadratic phase on matter waves, and counter-rotating orders are then brought to a common focus to form a multi-petal $l$0 intensity pattern (Lembessis, 2017). In acoustics, a wrapped leaky-wave metamaterial aperture produces integer and fractional vortex modes, including multi-core states, by exploiting dispersion-controlled effective topological order $l$1 (Naify et al., 2016).

5. Interaction-driven anomalies in scattering, reflection, and nonlinear media

A distinct class of anomalous vortex beams emerges when interactions convert vortex topology into new observables. In scattering theory, Bessel vortex states

$l$2

represent coherent superpositions of plane waves on a ring in transverse momentum space (Ivanov, 2012). In double-vortex scattering, a fixed final state can arise from two distinct initial momentum configurations, and the interference term depends on

$l$3

making the observable cross section sensitive to the phase difference of the underlying plane-wave amplitudes (Ivanov, 2012). The cited paper emphasizes that the phase-sensitive term scales linearly in the opening angle for non-zero OAM, whereas for non-vortex wave packets it is quadratically suppressed (Ivanov, 2012). This is anomalous from the plane-wave perspective because the process becomes its own interferometer.

Reflection from interfaces produces topological aberration. A high-order optical vortex reflected from a thin metallic film does not merely shift by Goos–Hänchen and Imbert–Fedorov amounts; its high-order core unfolds into a vortex constellation whose elementary symmetric polynomials encode first-, second-, and third-order aberrations (Barros et al., 2023). Near the attenuated-total-reflection plasmon resonance in a Kretschmann–Raether gold-film geometry, the barycenter of the vortex constellation shifts by more than 30 wavelengths, and higher-order deformations such as stretching and skewness are directly measured through the transformation of the vortex coordinates (Barros et al., 2023).

In electron microscopy, anomalousity is interaction-induced at atomic scales. When an electron vortex beam is centered on an atomic column in SrTiO$l$4, the beam exhibits significantly reduced delocalization compared with a conventional probe, while higher-order vortices split according to local site symmetry and form a three-dimensional vortex-line fabric with loops and twists around columns (Lubk et al., 2014). The topological structure rather than the OAM quantum number becomes the more robust organizing principle inside the crystal.

Nonlinear media introduce another anomaly: dissipation can stabilize rather than destroy vortex beams. Nonlinear Bessel vortex beams are stationary solutions of the nonlinear Schrödinger equation with multiphoton absorption,

$l$5

in which the inner nonlinear rings continuously transfer power and OAM to matter while being refueled by spiral inward currents from outer linear rings acting as an intrinsic reservoir (Porras et al., 2017). The cited stability analysis shows that some of these beams remain stable against azimuthal breakup precisely because of dissipation (Porras et al., 2017). This suggests a broader principle: anomalous vortex propagation can arise from non-conservative topological flux balance rather than conservative eigenmode structure.

A different kind of anomaly appears in beam models themselves. Within the complex source/sink model, exact nonparaxial vortex phasors with odd OAM exhibit analytic discontinuities at the beam waist for one common branch choice of the complex radius $l$6, producing nonphysical discontinuities in the electromagnetic field components. Choosing instead

$l$7

removes the discontinuity (Vikartofsky et al., 2017). This establishes that some purported exact vortex solutions are physically inadmissible unless branch choices are handled consistently.

6. Vector beams, intense-field regimes, and applications

Vector vortex beams constitute one of the most active subfamilies of anomalous vortex beams. In a vertical-cavity surface-emitting laser with frequency-selective feedback, anti-vortices with hyperbolic polarization structure, radially polarized vortices, four-domain vector vortices, and spiral vector vortices arise spontaneously as self-localized dissipative structures (Jimenez-Garcia et al., 2017). The anti-vortex is additionally a full Poincaré beam, with a spatially varying circular-polarization component measured through the Stokes parameter $l$8 (Jimenez-Garcia et al., 2017). The spontaneous emergence of these structures in a nearly isotropic VCSEL system indicates that anomalous vector vortex states need not be imposed externally by beam-shaping optics.

At high intensity, OAM reshapes plasma interaction physics. Intense vortex laser beams exhibit twisted phase fronts, hollow intensity distributions, spatially isolated longitudinal fields, and OAM delivery that modifies ponderomotive forces, wake topology, instabilities, and radiation (Shi et al., 2024). In plasma high-harmonic generation, the relativistic oscillating mirror model yields vortex harmonics with topological charge scaling $l$9, while in driven plasma waves the OAM-dependent dispersion relation

$2\pi l$0

governs twisted Langmuir modes (Shi et al., 2024). The review further identifies low-divergence particle acceleration, instability suppression, strong magnetic-field generation, and high-energy photon delivery with OAM as major application areas (Shi et al., 2024).

Modified radial structure adds another control parameter. Anomalous vortex (AV) and modified anomalous vortex (MAV) beams used to drive high-order harmonic generation are modeled by

$2\pi l$1

with $2\pi l$2 (Das et al., 10 Sep 2025). For MAV beams the radius of maximum intensity is

$2\pi l$3

allowing the near-field beam size to remain compact even at high topological charge and enabling XUV harmonics with topological charge approaching $2\pi l$4 while maintaining nearly uniform far-field divergence over harmonic orders 11–21 (Das et al., 10 Sep 2025). This suggests that anomalous radial engineering can decouple OAM from beam size in strong-field conversion.

Potential applications recur across the literature. Optical metrology, optical communications, and optical tweezers are proposed for Pearcey–Gauss-based anomalous beams (Rodríguez-Fajardo et al., 2024). High-resolution lithography, microscopy, and free-space communication are emphasized for metasurface-generated vector vortex beams (Yue et al., 2016). Hadronic phase measurements, meson photoproduction, and small-angle elastic scattering motivate vortex-beam collisions (Ivanov, 2012). Acoustic multiplexing, particle manipulation, and compact integrated emitters motivate fractional acoustic vortices (Naify et al., 2016). For composite matter waves, the cited outlook includes OAM-sensitive scattering, modified selection rules, and coupling to internal rotations or Rydberg states (Luski et al., 2021).

A recurring misconception is that anomalous vortex beams are defined only by “unusual” intensity shapes. The cited work suggests a broader and more precise view: anomaly may reside in caustic geometry, spin–orbit non-separability, symmetry-driven singularity splitting, space–time coupling, nonlinear stabilization, fractional topological staircases, or interaction-enabled observables. A plausible implication is that anomalous vortex beams are best understood not as exceptions to vortex optics but as its structurally richer generic case, once the assumptions of scalarity, cylindrical symmetry, monochromaticity, and weak interaction are relaxed.