Modified Anomalous Vortex Beams

• Modified anomalous vortex beams are crafted wavefields that deliberately depart from canonical vortex forms by engineering amplitude, phase, and symmetry.
• They employ techniques like symmetry-breaking apertures, radial amplitude engineering, and controlled phase discontinuities to achieve tailored OAM redistribution and singularity reorganization.
• These beams offer practical applications in HHG, electron microscopy, optical manipulation, and nano-focusing by maintaining compact size and enhanced performance at high topological charges.

Searching arXiv for recent and foundational papers on modified anomalous vortex beams and related vortex-beam engineering. Modified anomalous vortex beams are vortex-carrying wavefields whose amplitude, phase, symmetry, propagation, or generation geometry are deliberately altered so that their behavior departs from canonical circularly symmetric Laguerre–Gaussian or Bessel-type vortices while retaining, redistributing, or reinterpreting orbital angular momentum (OAM) and topological charge (TC). Across electron, optical, and structured-wave settings, the modification may take the form of symmetry-breaking apertures that induce OAM sidebands and vortex splitting, radial amplitude engineering that suppresses the usual OAM-dependent ring expansion, multi-ramp phase constructions that prescribe noncanonical TC jumps, analytically designed phase discontinuities that create robust non-circular topologies such as C-shaped beams, or engineered propagation schemes that lock a singularity to an accelerating main lobe (Clark et al., 2016, Das et al., 10 Sep 2025, Zeng et al., 2020, Mousley et al., 2016, Suzuki et al., 2021). Taken together, these works define a broad research area in which “anomalous” refers not to a single beam family but to controlled deviations from the standard vortex paradigm, including symmetry-driven OAM mixing, nontrivial singularity rearrangement, compact high-charge beams, propagation-invariant asymmetric structures, and multifocal or nano-focused topological fields (Rumi et al., 2017, Li et al., 2024, Yan et al., 2022).

1. Definitional scope and core invariants

A vortex beam is a paraxial wave with an azimuthal phase winding. For a single centered singularity, the field is locally written as $E \propto f(r)e^{i l \varphi}$, with integer $l$ the topological charge, and the corresponding OAM expectation for a pure mode is $\langle L_z\rangle=l\hbar$ per photon or $L_z=l\hbar$ for the electron-vortex eigenstate in cylindrically symmetric settings (Zeng et al., 2020, Clark et al., 2016). In electron microscopy, a pure circularly symmetric vortex eigenstate obeys $L_z\psi=\ell\hbar\psi$ with $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$ (Clark et al., 2016).

Modified anomalous vortex beams depart from this canonical case in distinct ways. In one line of work, “anomalous vortex beams” are beams whose topology and modal OAM content are modified by symmetry-breaking apertures, producing nontrivial splitting of a high-order on-axis vortex into multiple unit-charge vortices, vortex–antivortex pair creation, OAM sidebands, and far-field patterns that deviate from the pure-OAM ring structure (Clark et al., 2016). In another line, modified anomalous vortex beams (MAVBs) are engineered optical vortices with a tunable modification parameter $\delta$ and order $n$ that preserve high on-axis intensity and compact ring size at large topological charge, specifically for high-order harmonic generation (HHG) (Das et al., 10 Sep 2025). Elsewhere, the term naturally includes analytically tailored non-circular vortices such as C-shaped beams, multi-ramp fractional vortices with arbitrary TC-jump schedules, and vortex Airy beams whose embedded singularity is constrained to follow an accelerating lobe (Mousley et al., 2016, Zeng et al., 2020, Suzuki et al., 2021).

A central distinction throughout the literature is that TC and OAM need not behave identically. The net topological charge is a winding-count quantity,

$$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$

or equivalently a phase-winding integral on a large observation contour, whereas OAM is an expectation value computed from the azimuthal generator $i\partial_\varphi$ (Zeng et al., 2020). This distinction is crucial in fractional and multi-ramp constructions, where the OAM expectation varies smoothly while the net TC changes discontinuously according to engineered thresholds (Zeng et al., 2020). A similar separation appears in C-shaped beams: despite broken cylindrical symmetry, the beam carries a well-defined net OAM, $l$0, while its intensity distribution contains an adjustable macroscopic gap (Mousley et al., 2016).

This suggests that the unifying concept is not the violation of topological conservation, which is generally absent in linear propagation, but the controlled redistribution of singularities, modal content, and intensity morphology under constraints imposed by symmetry, apertures, discretization, nonlinearity, or propagation design (Clark et al., 2016, Rumi et al., 2017).

2. Mathematical frameworks for modification

Several mathematically distinct mechanisms generate modified anomalous behavior.

In symmetry-constrained electron-vortex propagation, the key object is the aperture function $l$1. For an aperture with $l$2-fold rotational symmetry,

$l$3

Multiplication of an input vortex $l$4 by $l$5 yields angular components $l$6, so an initially pure OAM state becomes a superposition of OAM sidebands differing by integer multiples of $l$7 (Clark et al., 2016). The post-aperture field is

$l$8

with coefficients determined by overlap integrals with the aperture geometry (Clark et al., 2016). This is the canonical symmetry-mixing mechanism.

In MAVB-driven HHG, the modification is radial and parametric rather than aperture-imposed. At the generation plane,

$l$9

with the modified radial coordinate

$\langle L_z\rangle=l\hbar$0

The special case $\langle L_z\rangle=l\hbar$1 is the anomalous vortex beam (AVB), while $\langle L_z\rangle=l\hbar$2 reduces to the zero-radial-index LG vortex (Das et al., 10 Sep 2025). The second-moment width and radius of maximum intensity,

$\langle L_z\rangle=l\hbar$3

show explicitly how $\langle L_z\rangle=l\hbar$4 and $\langle L_z\rangle=l\hbar$5 counteract the usual $\langle L_z\rangle=l\hbar$6-driven ring expansion (Das et al., 10 Sep 2025).

In analytically designed C-shaped beams, the modification is imposed via a phase mask over a circular aperture: $\langle L_z\rangle=l\hbar$7 The corresponding transmission,

$\langle L_z\rangle=l\hbar$8

contains both the conventional azimuthal term $\langle L_z\rangle=l\hbar$9 and a radially increasing spiral phase $L_z=l\hbar$0, which generates a controlled density of vortex–antivortex loops at the intended opening (Mousley et al., 2016). The beam is then obtained by Fraunhofer or Fresnel propagation of $L_z=l\hbar$1 (Mousley et al., 2016).

In anomalous multi-ramp fractional vortex (AMRFV) beams, the transmission function is piecewise defined over $L_z=l\hbar$2 azimuthal ramps: $L_z=l\hbar$3 Its Fourier expansion yields integer azimuthal orders with coefficients $L_z=l\hbar$4, so the propagated field is a superposition of integer-vortex components $L_z=l\hbar$5 weighted by $L_z=l\hbar$6 (Zeng et al., 2020). The net TC follows the additive law

$L_z=l\hbar$7

which allows arbitrary jump schedules through the choice of $L_z=l\hbar$8 (Zeng et al., 2020).

In discretized vortex-producing lenses (DVPLs), the continuous vortex-lens transmittance is quantized into $L_z=l\hbar$9 phase levels. The resulting field admits an azimuthal Fourier decomposition whose nonzero harmonics satisfy

$L_z\psi=\ell\hbar\psi$0

and whose coefficients carry sinc weighting and effective quadratic radial phase (Rumi et al., 2017). The propagated field is therefore a superposition of Kummer-type vortex beams of distinct effective charges and distinct focal planes (Rumi et al., 2017).

These formalisms are mutually different but structurally related: each replaces a pure $L_z\psi=\ell\hbar\psi$1 mode by a controlled superposition or deformation that redistributes the beam’s azimuthal spectrum, radial compactness, or singularity geometry (Clark et al., 2016, Das et al., 10 Sep 2025, Zeng et al., 2020, Rumi et al., 2017).

3. Symmetry, topology, and OAM redistribution

One of the most developed themes is the relation between discrete symmetry and topological restructuring. In electron-vortex propagation through apertures, the total topological charge enclosed by a contour is conserved under linear free-space propagation, but symmetry-breaking apertures redistribute singularities locally: a single on-axis charge $L_z\psi=\ell\hbar\psi$2 may split into several unit-charge vortices, while additional vortex–antivortex pairs may be created and annihilated without changing the net charge (Clark et al., 2016). The relevant global invariant is

$L_z\psi=\ell\hbar\psi$3

(Clark et al., 2016).

The Ferrando/García splitting rule governs when a high-order on-axis vortex tends to split under $L_z\psi=\ell\hbar\psi$4-fold symmetry. If $L_z\psi=\ell\hbar\psi$5, the central vortex tends to split; if $L_z\psi=\ell\hbar\psi$6, the central high-order vortex can remain undivided despite the broken cylindrical symmetry (Clark et al., 2016). This yields several representative cases. In a square aperture ($L_z\psi=\ell\hbar\psi$7), $L_z\psi=\ell\hbar\psi$8 satisfies $L_z\psi=\ell\hbar\psi$9, so the central vortex can remain undivided in the far field (Clark et al., 2016). In a triangular aperture ($\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$0), $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$1 both matches the symmetry and exceeds $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$2, so the central core splits into three unit-charge vortices arranged with triangular symmetry (Clark et al., 2016). For $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$3 in a square, splitting occurs and extra $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$4 pairs are required to satisfy both symmetry and charge conservation; the far-field central region then contains four $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$5 vortices and one $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$6 vortex, summing to $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$7 (Clark et al., 2016).

Discrete symmetry simultaneously determines OAM selection rules. A triangle ($\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$8) generates sidebands $\psi(r,\phi,z)\approx R(r,z)e^{i\ell\phi}$9, a square ($\delta$0) generates $\delta$1, while a centered circle preserves pure $\delta$2 and an off-centered circle, lacking rotational symmetry, induces general mixing across many $\delta$3 (Clark et al., 2016). This coupling between geometry and OAM content is central to the notion of anomalous modification.

A related but distinct selection-rule modification appears when optical vortex beams are incident obliquely on a Landau-quantized two-dimensional electron gas. Tilting the beam causes the longitudinal phase to expand in in-plane azimuthal harmonics weighted by Bessel functions $\delta$4, so a single beam-frame OAM $\delta$5 becomes a superposition of interface harmonics $\delta$6 (Takahashi et al., 2021). The laboratory-frame selection rule becomes

$\delta$7

rather than the normal-incidence locking $\delta$8 (Takahashi et al., 2021). This is not a beam-shaping study in the same sense as the others, but it shows that “modified anomalous” behavior can also emerge from geometry-induced OAM decomposition at an interface rather than from a phase mask or aperture (Takahashi et al., 2021).

A plausible implication is that the broader field treats symmetry breaking, coordinate deformation, and frame projection as formally analogous mechanisms: each converts a nominally pure vortex into a sideband comb or singularity network determined by the symmetry-breaking operator (Clark et al., 2016, Takahashi et al., 2021).

4. Propagation regimes, singularity dynamics, and robust asymmetric topologies

Propagation behavior separates modified anomalous beams into sharply different classes.

For symmetry-broken electron vortices, the near field immediately after the aperture is dominated by edge-wave interference. In triangle and square apertures, even $\delta$9 beams exhibit abundant $n$0 pairs due to Fresnel fringes from multiple edges; for $n$1 through a square aperture, simulations and experiments show a stable on-axis $n$2 core at the aperture plane, followed by rapid bifurcation into a complex vortex lattice with repeated creation–annihilation of $n$3 pairs forming three-dimensional vortex loops (Clark et al., 2016). As the Fresnel number approaches approximately 1, the system converges to the stable far-field arrangement dictated by symmetry and sideband structure (Clark et al., 2016).

By contrast, C-shaped vortex beams are designed specifically for propagation robustness. Their central vortex line co-propagates with a simple set of nodal lines, while peripheral vortex–antivortex loops sustain the opening of the “C” (Mousley et al., 2016). The beam undergoes slow rotation around focus attributed to Gouy phase, yet preserves the macroscopic C topology over defocus (Mousley et al., 2016). The robustness is not claimed as self-healing in the Bessel sense; rather, it arises from the central singularity and controlled singularity placement near the discontinuity (Mousley et al., 2016).

The same paper makes clear that the opening angle and size are independently tunable to leading order: the size $n$4 scales mainly with $n$5, while the opening angle $n$6 is governed chiefly by $n$7, through the density and arrangement of vortex–antivortex loops (Mousley et al., 2016). The gap is therefore not a missing segment of a ring but a topologically stabilized dark sector created by dislocation packing (Mousley et al., 2016).

The new-type vortex Airy beam introduces a different propagation constraint: a charge-$n$8 singularity remains locked to the accelerating main lobe because the field is constructed as

$n$9

using laterally sheared Airy beams with a $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$0 phase shift (Suzuki et al., 2021). In the ideal case $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$1, the envelope translates rigidly along the Airy trajectory $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$2, and the singularity follows it (Suzuki et al., 2021). For finite-energy truncation $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$3, the deviation remains small for $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$4, and for $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$5 it is substantially smaller than in conventional vortex Airy beams (Suzuki et al., 2021). The inner-lobe OAM mode purity in $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$6 reaches 91% for the new-type beam versus 81% for the conventional construction, an improvement up to 10% (Suzuki et al., 2021).

A distinct propagation anomaly occurs in DVPL-generated beams, where discretization produces multi-focal behavior. Because only harmonics $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$7 survive and each carries an effective quadratic phase, the focal plane of each component is

$$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$8

so distinct topological charges preferentially form at distinct longitudinal planes (Rumi et al., 2017). The propagated field is therefore a sum of Kummer-type beams with different effective charges and focal positions, while the total topological charge remains conserved (Rumi et al., 2017). This is a different kind of anomalous propagation: not local vortex splitting but axial stratification of modal orders.

5. Beam-size control, perfection, and compact high-charge beams

A major motivation for beam modification is the suppression of the standard increase of vortex-ring radius with topological charge.

In HHG, conventional LG drivers with zero radial index have ring radius $$Q = \frac{1}{2\pi}\oint_C \nabla \arg E \cdot d\boldsymbol{\ell},$$9, which reduces peak intensity and hinders nonlinear conversion at high $i\partial_\varphi$0 (Das et al., 10 Sep 2025). MAVBs address this by introducing the $i\partial_\varphi$1-$i\partial_\varphi$2 radial modification described above. Increasing $i\partial_\varphi$3 at fixed $i\partial_\varphi$4 reduces beam size and narrows the ring; increasing $i\partial_\varphi$5 further suppresses the $i\partial_\varphi$6-driven ring expansion (Das et al., 10 Sep 2025). The far-field harmonics preserve the OAM rule

$i\partial_\varphi$7

verified by counting $i\partial_\varphi$8 phase wraps in the harmonics (Das et al., 10 Sep 2025). The reported examples include a driver with $i\partial_\varphi$9 producing $l$00 at the 17th harmonic and $l$01 at the 21st harmonic, while maintaining compact ring size and nearly uniform divergence across harmonics 11th–21st (Das et al., 10 Sep 2025). The driver wavelength is $l$02 nm, with intensities $l$03–$l$04 W/cm$l$05, and short quantum trajectories within a thin-slab model (Das et al., 10 Sep 2025).

Laguerre–Gaussian Perfect Vortex Beams (LGPVBs) solve the same radius-inflation problem from a different direction. Starting from an analytic expression for the main-lobe radius of an LG beam, the radial index $l$06 is chosen as a function of OAM $l$07 so that all modes share the same waist-plane ring radius $l$08 (Yan et al., 2022). With fixed $l$09, the construction yields

$l$10

so the ring radius is OAM-independent at all $l$11, not merely at a focal plane (Yan et al., 2022). The beam retains LG self-similarity and supports arbitrary longitudinal intensity shaping and self-acceleration through angular-spectrum design (Yan et al., 2022). Experimentally demonstrated cases include $l$12 mm, $l$13 mm, and $l$14, all showing the same ring radius at $l$15, 413, and 826 mm (Yan et al., 2022).

The relationship between MAVBs and LGPVBs is conceptual rather than identical. MAVBs use a modified radial coordinate to maintain compact high-charge beams in a nonlinear generation context (Das et al., 10 Sep 2025), whereas LGPVBs choose the LG radial order $l$16 to enforce three-dimensional “perfectness” across propagation (Yan et al., 2022). This suggests two design philosophies for anomalous compactness: amplitude–phase self-focusing in the source plane, or modal-order compensation across the propagation law.

At the nanoscale, a further form of compactification is achieved by hyperbolic metamaterials. A circularly polarized input beam is converted predominantly into a radially polarized vortex with $l$17 by a hypergrating, and the high-$l$18 modes of the hyperbolic structure compress the focal feature to a simulated FWHM of approximately 110 nm at $l$19 nm, about $l$20, with experiments showing a feature around 200 nm diameter, about $l$21 (Li et al., 2024). The near field exhibits Néel-type skyrmion-like spin textures, with spin flips on 8 nm and 2 nm scales in the first and second domains respectively (Li et al., 2024). The output is a coherent mixture of a radially polarized OAM component and residual circularly polarized components with measured charges $l$22 and $l$23 (Li et al., 2024). Here the modification is neither perfect-vortex engineering nor standard anomalous radial shaping, but sub-diffraction nano-focusing plus SAM–OAM conversion in an anisotropic medium (Li et al., 2024).

6. Generation architectures, diagnostics, and representative implementations

The literature uses a wide range of platforms, reflecting the fact that modified anomalous vortex beams are a design family rather than a single apparatus class.

In transmission electron microscopy, forked holographic masks generate electron vortices with $l$24, while geometric apertures placed in the selected-area plane impose symmetry or asymmetry (Clark et al., 2016). The reported experiments used an aberration-corrected FEI Titan$l$25 TEM at 200 keV for far-field measurements and 300 keV for propagation series, with a 10 $l$26m forked holographic mask and 2 $l$27m characteristic-size apertures projected to about 30 nm (Clark et al., 2016). Circle, triangle, square, and off-centered circle apertures were tested (Clark et al., 2016).

For optical C-shaped electron beams, the analytical phase was encoded in a binary computer-generated hologram fabricated by focused ion beam milling on a 200 nm silicon nitride membrane initially coated with about 50 nm Pt/Pd (Mousley et al., 2016). The target example used $l$28 and $l$29, corresponding to an opening angle $l$30 and size $l$31, and was measured in a TEM at 200 kV (Mousley et al., 2016). The measured far-field pattern closely matched simulation, with some intensity asymmetry attributed to milling imperfections (Mousley et al., 2016).

MAVBs for HHG are presented as simulations, but the generation route is explicit: spatial light modulators can implement a composite amplitude–phase hologram encoding the helical phase $l$32 and the radial amplitude $l$33 (Das et al., 10 Sep 2025). The gas target is placed at focus $l$34, and the harmonics are analyzed in the far field via thin-slab modeling and Fraunhofer diffraction (Das et al., 10 Sep 2025).

LGPVBs were generated experimentally with a phase-only SLM (Holoeye GAEA-2, pixel pitch 3.7 $l$35m, 3840$l$362160), a 4$l$37 filtering stage, and a Fourier lens of focal length 200 mm (Yan et al., 2022). A relay system and delay line recorded cross-sections at different $l$38, allowing direct verification of 3D perfection, self-healing, and shaped propagation (Yan et al., 2022).

New-type vortex Airy beams were generated using a Ti:sapphire regenerative amplifier at 800 nm, a phase-only SLM encoding the composite Fourier-domain phase, and a 300 mm Fourier lens (Suzuki et al., 2021). Interference with a reference beam revealed the two-pronged fork associated with the embedded $l$39 singularity (Suzuki et al., 2021).

DVPL-based anomalous vortices were implemented on a Holoeye LC2002 twisted-nematic liquid-crystal SLM with maximum phase stroke about $l$40 and about 5% amplitude coupling (Rumi et al., 2017). The experiments used $l$41 nm, $l$42 mm, $l$43 cm, and $l$44 m (Rumi et al., 2017). The observed focal planes and topological charges matched the predicted multifocal structure (Rumi et al., 2017).

In hyperbolic metamaterial nano-focusing, the platform consists of alternating Ag and Ti$l$45O$l$46 layers of nominally 30 nm thickness, with Fresnel zones milled into approximately 50 nm Cr and planarized by PMMA before deposition of the multilayer (Li et al., 2024). At 532 nm, the effective-medium parameters are $l$47 and $l$48, i.e. a type-II hyperbolic metamaterial (Li et al., 2024). Stokes polarimetry showed the OAM radial component carried $l$49 of the total intensity, with about 13% circular and about 11% unpolarized light attributed to scattering (Li et al., 2024).

7. Applications, limitations, and open directions

The application space is correspondingly heterogeneous. In electron microscopy, isolating symmetry-driven OAM mixing clarifies mechanisms relevant to vortex-EMCD optimization and to understanding how beam topology responds to sample or aperture symmetries (Clark et al., 2016). In HHG, MAVBs enable XUV harmonics with large topological charge, compact size, and nearly uniform divergence, which the paper associates with high-resolution microscopy, nanoscale chiral spectroscopy, spin–orbit-sensitive studies, OAM-based metrology, and XUV beam shaping (Das et al., 10 Sep 2025). C-shaped vortices are proposed for lithography, dynamical atom sorting, atomtronics, and field sensing (Mousley et al., 2016). LGPVBs target OAM multiplexing, fiber coupling, optical manipulation, and propagation in lossy or turbid media (Yan et al., 2022). Hyperbolic metamaterial focusing is positioned toward enhanced chiral and forbidden transitions, OAM spectroscopy, nanoscale optomechanics, data storage, and quantum light–matter interaction experiments (Li et al., 2024).

Several objective limitations recur. Many analyses rely on paraxial propagation, including the electron-aperture, Airy, LG-perfect, and HHG beam formulations (Clark et al., 2016, Suzuki et al., 2021, Yan et al., 2022, Das et al., 10 Sep 2025). In HHG, the thin-slab model omits full macroscopic phase matching and gas propagation (Das et al., 10 Sep 2025). In electron beams, multiple scattering, channelling, and crystal symmetry inside matter may impose additional OAM mixing beyond the isolated aperture effects (Clark et al., 2016). In C-shaped beams, the mapping from $l$50 to $l$51 is empirical rather than given by a closed scaling law (Mousley et al., 2016). In AMRFV beams, the half-integer jump law is derived for wide beams, whereas finite-width beams shift thresholds toward integer effective charges (Zeng et al., 2020). In DVPLs, discretization creates undesired sidebands that can be suppressed but not eliminated except in limiting regimes; low phase-stroke SLMs restrict usable discretization levels (Rumi et al., 2017). In hyperbolic metamaterials, losses, roughness, EMT deviations for approximately 30 nm layers, and the evanescent nature of the focus limit direct outcoupling and reduce modal purity (Li et al., 2024).

A common misconception is that anomalous or modified vortices abandon topological conservation. The surveyed works do not support that view. Instead, they show that global topological charge is conserved while local singularity content, OAM spectrum, or spatial morphology can change dramatically under propagation or engineered transformations (Clark et al., 2016, Rumi et al., 2017). Another misconception is that broken cylindrical symmetry necessarily destroys meaningful OAM; in fact, C-shaped beams retain a narrow and well-defined OAM content under analytical design, and symmetry-broken electron beams can retain a dominant modal component even when the phase map contains multiple singularities, as in the square-aperture $l$52 case where approximately 86% of the wavefront remains in the $l$53 component (Mousley et al., 2016, Clark et al., 2016).

The field’s broader significance lies in treating the optical or electron vortex not as a fixed doughnut mode but as a programmable topological resource. Depending on the transformation, one can prescribe sideband spacing through aperture symmetry, compactify high-charge drivers through radial self-focusing, impose arbitrary TC jump schedules through multi-ramp phase plates, embed robust non-circular gaps through analytically organized vortex–antivortex loops, stratify different effective charges along the optical axis via discretized vortex lenses, lock singularities to accelerating lobes, or compress vortex structure to deep-subwavelength scales with spin-textured near fields (Clark et al., 2016, Das et al., 10 Sep 2025, Zeng et al., 2020, Mousley et al., 2016, Rumi et al., 2017, Suzuki et al., 2021, Li et al., 2024). This suggests that “modified anomalous vortex beams” is best understood as a general category of deliberately engineered topological beams whose departures from canonical symmetry are not defects but design variables.