Pith. sign in

REVIEW 3 major objections 5 minor 40 references

Orbital angular momentum of spatiotemporal vortices: a ray-mechanical analogy

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A particle-loop model with non-uniform mass and a semiclassical vorticity condition reproduces the previously conflicting transverse OAM values of spatiotemporal vortex pulses.

desk verdict A clean mechanical model that traces the conflicting STVP OAM values to centroid conventions, but its exact reproduction of prior results hinges on an unproven Fourier-space ellipse relation. read the letter →

arxiv 2601.15261 v1 pith:SXTBOSLF submitted 2026-01-21 physics.optics

Sophie Vo , Konstantin Y. Bliokh , Miguel A. Alonso This is my paper · ORCID
classification physics.optics
keywords spatiotemporalvortexpulsestransverseorbitalangularmomentumrayopticsmechanicalparticleloopvorticityquantizationenergycentroiddispersionrelation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the long-standing discrepancies in the transverse orbital angular momentum (OAM) of spatiotemporal vortex pulses (STVPs) are not fundamental, but follow from the choice of reference point and wave dispersion. The authors build a minimal mechanical model: a loop of non-interacting particles moving at constant speed along rays, with a mass density chosen to mimic the pulse's momentum spectrum. Supplemented by a semiclassical condition that the phase accumulated around the loop equals 2π times the topological charge, the model yields exactly the wave-based OAM expressions from previous work — γℓ/2 at the energy centroid, (γ+γ⁻¹)ℓ/2 at the particle centroid, and -ℓ/(2γ) at the point minimizing the squared OAM. The model also shows why uniform-mass particle loops give different answers and why dispersion changes the intrinsic OAM. For a reader, the payoff is a single intuitive picture that reconciles apparently contradictory results and explains where each formula comes from.

What carries the argument

The central object is a ray-mechanical particle loop: a family of non-interacting point particles indexed by ξ∈[0,2π), each moving along a rectilinear ray at speed c with direction u(ξ)≃(Δx cosξ, 1). The loop's initial conditions set the ellipse to be spatial (xz STVP) or spatiotemporal (xt STVP). The key refinement is a non-uniform mass distribution p(ξ)=mc/2π(Δx cosξ, 1-Δz sinξ), chosen to mimic the elliptical distribution of the wavevector spectrum. The argument is carried by the semiclassical phase Φ(ξ) satisfying ∂Φ/∂ξ=k(ξ)·∂r/∂ξ, with k(ξ)=2πp(ξ), and the quantization condition Φ(2π)=2πℓ, which in the paraxial regime becomes k0(Δx wx + Δz wz,t)=2ℓ. Dividing by the inverse-ellipse relat

What would settle it

Measure or compute the transverse OAM at the energy centroid for a paraxial STVP whose spectral ellipse axes do not obey Δz/Δx = 1/γ (e.g., an STVP engineered with independent control of spatial and spectral widths); if the result diverges from γℓ/2, the model's reproduction is contingent on that relation.

Watch

Extended reading notes

Core claim

The central discovery is that the mechanical particle-loop model, when the particles carry a non-uniform mass density proportional to the wave spectrum and when the phase around the loop is quantized, reproduces the previously reported transverse OAM of STVPs in both the xz and xt frameworks. In detail, the OAM with respect to the mass (energy) centroid is γℓ/2, with respect to the particle (number-density) centroid is (γ+γ⁻¹)ℓ/2, and with respect to the point minimizing the OAM magnitude is -ℓ/(2γ), where ℓ is the topological charge and γ is the ratio of the longitudinal and transverse ellipse semiaxes. These match the wave-based results previously reported. The model traces the apparent co

Load-bearing premise

The derivation relies on the asserted relation Δz/Δx = 1/γ between the momentum- and real-space ellipse axes; if a real STVP's spectrum does not obey it, the reproduced OAM values change.

Editorial extensions

If this is right

  • The transverse OAM of an STVP is not a single number: different reference points (energy centroid vs particle centroid) give different intrinsic values, and this explains the reported spread of formulas.
  • For circular STVPs (γ=1), the model predicts half-integer OAM ℓ/2 at the energy centroid and integer OAM ℓ at the particle centroid.
  • The same model applies to any wave with linear dispersion (light, sound); for nonrelativistic quantum particles the mass and particle centroids coincide and the intrinsic OAM becomes (γ+γ⁻¹)ℓ/2.
  • The quantization condition shows that the spatial width needed for a given topological charge is twice as large in the uniform-mass model as in the non-uniform one — a prediction that could be checked in wavefields.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the inverse-ellipse relation is not exactly satisfied by real STVPs, the model suggests that the three published OAM values sit on a continuum parameterized by the spectral ellipticity, so experiments measuring OAM could in principle distinguish which centroid definition the pulse actually realizes.
  • The ray-loop picture could be extended to fractional vorticity: when Φ(2π) is not an integer multiple of 2π, the wavefield shows an intensity cut that could serve as a mechanical model of fractional STVPs, with the interruption position dependent on the integration range.
  • A natural testable extrapolation: the model predicts that for non-paraxial STVPs the discrepancy between centroids and OAM values grows, with expressions in terms of complete elliptic integrals; comparing those with a full nonparaxial wave calculation would test whether the mechanical analogy holds beyond the paraxial regime.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a mechanical analogy for spatiotemporal vortex pulses (STVPs), modeling them as a loop of independent point particles moving at speed c along rays. It examines two frameworks—xz-elliptic and xt-elliptic initial conditions—and two mass distributions: uniform and non-uniform. With the non-uniform mass density chosen to mimic the STVP wavevector spectrum, and after imposing a semiclassical vorticity quantization condition, the model yields transverse OAM expressions (Table V) that the authors claim exactly reproduce previously reported wave-based results (Table I). The paper also provides Gaussian-dressed wave estimates, an exact nonparaxial appendix, and a discussion of dispersion. The algebraic derivations are clean, but the central claim depends on an unproven inverse-ellipse relation between real- and momentum-space axes.

Significance. If the claim is fully established, the model would be a valuable conceptual tool: it explains in simple ray/particle terms why different centroid prescriptions yield different OAM values, why the xz and xt frameworks give equivalent paraxial results, and how nonrelativistic dispersion changes the intrinsic OAM. The explicit tables, nonparaxial closed forms, and wave-dressed visualizations are useful strengths. However, the 'exact reproduction' is weaker than stated: the non-uniform momentum distribution is imported from one previous wave calculation, the quantization condition is imposed, and the numerical match to Table V hinges on an unproven relation. The model nevertheless makes a falsifiable prediction for nonrelativistic STVPs, which is a plus.

major comments (3)
  1. [III.C, Eq. (19)] The inverse-ellipse relation Δz/Δx = 1/γ is asserted ('should also correspond') without derivation. It is load-bearing: with r = Δz/Δx, Eq. (19) gives k0(1+γr)Δx w_x = 2ℓ, and the Table IV OAM expressions become ⟨Ly⟩(xmass)=γℓ/(1+γr), ⟨Ly⟩(xpart)=(r+γ)ℓ/(1+γr), ⟨Ly⟩(xmin)=−rℓ/(1+γr). These reduce to Table V only for r=1/γ. The ratio must be derived from the STVP spectrum or introduced as an explicit assumption, and the 'exact reproduction' claim must be qualified accordingly.
  2. [III.A and III.C] The non-uniform mass density Eq. (13) is chosen to mimic the wavevector spectrum of Ref. [15], and the quantization condition Eq. (19) is imposed to define ℓ. The reproduced OAM values therefore follow partly by construction. The paper should clearly separate inputs from outputs and identify which results are genuine predictions—e.g., the factor 1/2 relative to the uniform-mass model, the centroid differences, and the dispersion dependence—rather than presenting the match to Table I as an independent confirmation.
  3. [IV (concluding remarks)] The claim to 'reproduce the results of previous wave-based calculations [14,15,17,20,29,30]' overstates Table V. Table V contains only the three expressions γℓ/2, (γ+γ^{-1})ℓ/2, and −ℓ/(2γ). It does not include Porras's Ly=0 or LEXT_y=−γ/2ℓ from Table I. The concluding claim should list exactly which table entries are matched and which are not.
minor comments (5)
  1. [II.D, Eq. (12)] Typo: the phase term k0{[x−X(ξ,0)]u_x(ξ)+[x−Z(ξ,0)]u_z(ξ)} should have [z−Z(ξ,0)]u_z(ξ).
  2. [III.B, Eq. (15)] The first OAM expression is ambiguous: it should be mc(x0 + Δz wx/2 + Δx wz/2) to match Table IV; as printed, evaluation at xmass = −Δz wx/2 does not give the Table IV value.
  3. [III.C, Eq. (19)] The notation 'Δz wz,t' is unclear; specify that it is Δz wz for xz STVPs and Δz wt for xt STVPs.
  4. [III.C] The statement 'precisely reproduce the results obtained in [14–16,20]' does not match Table I, which lists [14,15,17,20]; check the reference mapping.
  5. [Appendix A] In Tables VI and VII, the relationship between x0min and the xmin of the main text is not explicitly stated; please clarify the notation. Also 'Supplementary Document A' should be 'Supplemental Document A'.

Circularity Check

2 steps flagged · score 6.0 of 10

Exact reproduction of Table I hinges on the unproven inverse-ellipse relation Δz/Δx = 1/γ and on importing Ref. [15]'s wave spectrum, so the central match is partly constructed.

  1. fitted input called prediction [Sec. III.A, Eq. (13); Abstract]
    "we consider a ray family with momentum distribution p(ξ) mimicking the elliptical wavevector distribution in [15] ... Remarkably, when supplemented by a semiclassical vorticity quantization condition, our mechanical model exactly reproduces different wave-based OAM results previously reported for paraxial STVPs."

    The momentum-space input p(ξ) is taken from the very prior wave calculation [15] whose OAM values are the claimed output. The mechanical OAM is a first moment of this imported position-momentum distribution, so the agreement of Table V with Table I is partly built into the input; the model adds centroid choices and a quantization condition, but the numerical spectrum is adopted, not independently derived.

  2. self definitional [Sec. III.C, after Eq. (19), before Table V]
    "Similarly to Section II, we introduce the ratio of the STVP ellipse semiaxes in real space, γ=w_z/w_x = w_t/w_x, which should also correspond to the inverted relation for the momentum-space ellipse in Eq. (13): Δz/Δx = 1/γ."

    With r=Δz/Δx, Eq. (19) becomes k0(1+γr)Δx w_x = 2ℓ, and Table IV gives ⟨Ly⟩(x_mass)=γℓ/(1+γr), ⟨Ly⟩(x_part)=(r+γ)ℓ/(1+γr), ⟨Ly⟩(x_min)=-rℓ/(1+γr). These equal the Table I values γℓ/2, (γ+γ⁻¹)ℓ/2, and −ℓ/(2γ) only for r=1/γ, which is exactly the relation asserted without derivation. Thus the claimed exact reproduction is imposed by this unproven correspondence, not produced by the mechanical model alone.

full rationale

The model is not wholly circular: the uniform-mass particle-loop stage produces γℓ and 0, which match no prior wave result, indicating independent mechanical content. However, the non-uniform-mass stage imports the wave spectrum of Ref. [15] (Eq. 13) and then imposes the inverse-ellipse relation Δz/Δx=1/γ without deriving it. The algebra shows Table V collapses to Table I precisely when that relation holds; absent it, the mechanical OAM values form a one-parameter family and the exact reproduction fails. Because the critical relation is asserted rather than derived and the spectrum is the same prior work whose OAM values are the benchmark, the 'remarkable reproduction' is in part constructed rather than predicted. This is partial circularity (6), not complete: alternative mass distributions give different results and the centroid logic is independently motivated. A separate correctness concern is that the concluding claim to reproduce [14,15,17,20,29,30] overstates Table V, which does not reproduce Porras's Ly=0 or LEXT=−γ/2ℓ from Table I.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the quantization condition and on importing the wave spectrum into the mechanical model; the paper's own uniform-mass variant gives different OAM (γℓ), showing that the non-uniform mass density is essential to the reproduction.

free parameters (2)
  • Δz/Δx inverse-ellipse ratio = 1/γ (Δz = Δx w_x/w_z,t)
    Assumed in Section III.C to convert the quantization condition (19) into k0 Δx w_x = ℓ; without it the OAM values would not match Table I. This is a modeling choice, not derived in the paper.
  • w_dress (Gaussian dressing width) = sqrt(w_x/(k0 Δx))
    Chosen in Eq. (12) to render the wave estimates in Figs. 4, 5, and 8; not part of the OAM derivation.
assumptions (5)
  • domain assumption A STVP can be represented by a loop of non-interacting particles moving at speed c along rectilinear rays (Eq. 1).
    Borrowed from Ref. [18]; the foundation of the mechanical analogy.
  • domain assumption Phase consistency along the closed ray loop requires Φ(2π) = 2πℓ with ℓ integer (semiclassical Bohr-Sommerfeld quantization, Eqs. 10 and 18).
    Connects the classical particle OAM to the wave topological charge; not derived from Maxwell's equations.
  • ad hoc to paper The non-uniform mass density is chosen so that the particle momenta (Eq. 13) mimic the elliptical wavevector spectrum of the STVP from Ref. [15].
    The mechanical model is calibrated to the wave spectrum it aims to reproduce; this is the main reverse-engineering step.
  • domain assumption The transverse OAM of the wave equals the mechanical OAM of the particle loop computed with Eq. (6).
    Paraxial ray/wave correspondence underlying all OAM results.
  • domain assumption Optical and sound waves obey a linear dispersion relation E = pc, so the mass density can be taken proportional to the momentum magnitude (Section III.E).
    Justifies the mapping |p(ξ)| ∝ mass density; nonlinear dispersion would change the centroids and OAM.
invented entities (1)
  • Non-uniform mass density m(ξ) along the particle loop
    purpose: To mimic the energy/momentum distribution of the wave spectrum so the particle model reproduces wave OAM values
    A modeling device rather than a physical field; it encodes the photon dispersion relation into the classical model and is chosen specifically to match Ref. [15].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Orbital angular momentum of spatiotemporal vortices: a ray-mechanical analogy." pith.science (2026). https://pith.science/paper/SXTBOSLF

@misc{pith2026260115261,
  author       = {Pith},
  title        = {Pith review of: Orbital angular momentum of spatiotemporal vortices: a ray-mechanical analogy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXTBOSLF}},
  note         = {Machine review of arXiv:2601.15261}
}
read the original abstract

Spatiotemporal vortex pulses (STVPs) are wavepackets that carry transverse orbital angular momentum (OAM), whose proper quantification has been the subject of recent debate. In this work, we introduce a simplified mechanical model of STVPs, consisting of a loop of non-interacting point particles traveling at a uniform constant speed but at slightly di!erent angles. We examine di!erent initial conditions for the particle loop, including configurations that are elliptic in space at a given time and configurations that are elliptic in spacetime at a fixed propagation distance. Furthermore, employing a non-uniform mass distribution allows the particle loop to mimic the STVP not only in configuration space but also in momentum space. Remarkably, when supplemented by a semiclassical vorticity quantization condition, our mechanical model exactly reproduces di!erent wave-based OAM results previously reported for paraxial STVPs.

Figures

Figures reproduced from arXiv: 2601.15261 by the authors.

Figure 2
Figure 2. FIG. 2. Ray-optics modeling of [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Rays corresponding to trajectories of particles for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Ray-optics modeling of [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Intensity and real part of the wavefield [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Effect of the non-satisfaction of the quantization [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Ray-optics modeling of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Ray-optics modeling of [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Same as in Fig. 4 but for the model with a non [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 1 canonical work pages

  1. [15]

    K. Y. Bliokh, Orbital angular momentum of optical, acoustic, and quantum-mechanical spatiotemporal vor- tex pulses, Phys. Rev. A107, L031501 (2023)

    2023

  2. [1]

    M. S. Soskin and M. V. Vasnetsov, Singular optics, Prog. Opt.42, 219 (2001)

    2001

  3. [2]

    M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singu- lar optics: Optical vortices and polarization singularities, Prog. Opt.53, 293 (2009)

    2009

  4. [3]

    K. Y. Bliokh, I. P. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. Béché, R. Juchtmans, M. A. Alonso, P. Schattschneider, F. Nori, and J. Verbeeck, Theory andapplicationsoffree-electronvortexstates,Phys.Rep. 690, 1 (2017)

    2017

  5. [4]

    J. F. Nye and M. V. Berry, Dislocations in wave trains, Proc. R. Soc. Lond. A336, 165 (1974). 9

    1974

  6. [5]

    Allen, S

    L. Allen, S. M. Barnett, and M. J. Padgett,Optical An- gular Momentum(IoP Publishing, 2003)

    2003

  7. [6]

    D. L. Andrews and M. Babiker,The Angular Momen- tum of Light(Cambridge University Press, Cambridge, England, UK, 2012)

    2012

  8. [7]

    K. Y. Bliokh and F. Nori, Transverse and longitudinal angular momenta of light, Phys. Rep.592, 1 (2015)

    2015

Show all 40 references
  1. [8]

    K. Y. Bliokh, Momentum, spin, and orbital angular mo- mentum of electromagnetic, acoustic, and water waves, Contemp. Phys.65, 219 (2024)

    2024

  2. [9]

    Allen, M

    L. Allen, M. W. Beijersbergen, R. J. C. Spereeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A45, 8185 (1992)

    1992

  3. [10]

    P. H. Ceperley, Rotating waves, Am. J. Phys.60, 938 (1992)

    1992

  4. [11]

    M. V. Berry, Paraxial beams of spinning light, Proc. of SPIE3487, 6 (1998)

    1998

  5. [12]

    A. P. Sukhorukov and V. V. Yangirova, Spatio-temporal vortices: properties, generation and recording, Proc. SPIE5949, 594906 (2005)

    2005

  6. [13]

    K. Y. Bliokh and F. Nori, Spatiotemporal vortex beams and angular momentum, Phys. Rev. A86,033824 (2012)

    2012

  7. [14]

    S. W. Hancock, S. Zahedpour, and H. M. Milchberg, Mode structure and orbital angular momentum of spa- tiotemporal optical vortex pulses, Phys. Rev. Lett.127, 193901 (2021)

    2021

  8. [16]

    M. A. Porras, Propagation of higher-order spatiotempo- ral vortices, Opt. Lett.48, 367 (2023)

    2023

  9. [17]

    M. A. Porras, Transverse orbital angular momentum of spatiotemporal optical vortices, Prog. Electromagn. Res. 177, 95 (2023)

    2023

  10. [18]

    S. Vo, R. Gutiérrez-Cuevas, and M. A. Alonso, Closed forms for spatiotemporal optical vortices and sagittal skyrmionic pulses, J. Opt.26, 95607 (2024)

    2024

  11. [19]

    H. Wang, C. Guo, W. Jin, A. Y. Song, and S. Fan, En- gineering arbitrarily oriented spatiotemporal optical vor- tices using transmission nodal lines, Optica8, 966 (2021)

    2021

  12. [20]

    Bekshaev, Spatiotemporal optical vortices: principles of description and basic properties, APL Photonics9, 110806 (2024)

    A. Bekshaev, Spatiotemporal optical vortices: principles of description and basic properties, APL Photonics9, 110806 (2024)

    2024

  13. [21]

    Jhajj, I

    N. Jhajj, I. Larkin, E. W. Rosenthal, S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, Spatiotemporal opti- cal vortices, Phys. Rev. X6, 031037 (2016)

    2016

  14. [22]

    Chong, C

    A. Chong, C. Wan, J. Chen, and Q. Zhan, Generation of spatiotemporal optical vortices with controllable trans- verse orbital angular momentum, Nat. Photon.14, 350 (2020)

    2020

  15. [23]

    S. W. Hancock, S. Zahedpour, A. Goffin, and H. M. Milchberg, Free-space propagation of spatiotemporal op- tical vortices, Optica6, 1547 (2019)

    2019

  16. [24]

    Y. Zang, A. Mirando, and A. Chong, Spatiotemporal op- tical vortices with arbitrary orbital angular momentum orientation by astigmatic mode converters, Nanophoton- ics11, 745 (2022)

    2022

  17. [25]

    Martín-Hernández, G

    R. Martín-Hernández, G. Gui, L. Plaja, H. C. Kapteyn, M. M. Murnane, C.-T. Liao, M. A. Porras, and C. Hernández-García, Extreme-ultraviolet spatiotempo- ral vortices via high harmonic generation, Nat. Photonics 19, 817 (2025)

    2025

  18. [26]

    Zhang, Y

    H. Zhang, Y. Sun, J. Huang, B. Wu, Z. Yang, K. Y. Bliokh, and Z. Ruan, Topologically crafted spatiotempo- ral vortices in acoustics, Nat. Commun.14, 6238 (2023)

    2023

  19. [27]

    H. Ge, S. Liu, X.-Y. Xu, Z.-W. Long, Y. Tian, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, Spatiotemporal Acoustic Vor- tex Beams with Transverse Orbital Angular Momentum, Phys. Rev. Lett.131, 014001 (2023)

    2023

  20. [28]

    Z. Che, W. Liu, J. Ye, L. Shi, C. T. Chan, and J. Zi, Generation of Spatiotemporal Vortex Pulses by Resonant Diffractive Grating, Phys. Rev. Lett.132, 044001 (2024)

    2024

  21. [29]

    M. A. Porras, Clarification of the transverse orbital an- gular momentum of spatiotemporal optical vortices, J. Opt.26, 095601 (2024)

    2024

  22. [30]

    Tripathi, S

    N. Tripathi, S. W. Hancock, and H. M. Milchberg, Trans- verse orbital angular momentum of spatiotemporal opti- cal vortices: setting the record straight, Opt. Express33, 41766 (2025)

    2025

  23. [31]

    Gadeyne and T

    T. Gadeyne and T. Ruchon, Energy and photon centroids of spatiotemporal light pulses and consequences for their intrinsic orbital angular momentum, Phys. Rev. A112, 043519 (2025)

    2025

  24. [32]

    M. A. Alonso and G. W. Forbes, Uncertainty products for nonparaxial wave fields, J. Opt. Soc. Am. A17, 2391 (2000)

    2000

  25. [33]

    Y. A. Kravtsov, Two new asymptotic methods in the the- ory of wave propagation in inhomogeneous media, Sov. Phys. Acoust.14, 1 (1968)

    1968

  26. [34]

    M. V. Berry and K. E. Mount, Semiclassical approxima- tionsinwavemechanics,Rep.Prog.Phys.35,315(1972)

    1972

  27. [35]

    Huang, P

    S. Huang, P. Wang, Y. Xu, J. Liu, and R. Li, Frac- tional spatiotemporal optical vortices, arXiv:2504.11697 10.48550/arXiv.2504.11697 (2025)

    2025 doi arXiv

  28. [36]

    M. V. Berry, Optical vortices evolving from helicoidal integer and fractional phase steps, J. Opt. A: Pure Appl. Opt.6, 259 (2004)

    2004

  29. [37]

    Leach, E

    J. Leach, E. Yao, and M. J. Padgett, Observation of the vortex structure of a non-integer vortex beam, New J. Phys.6, 71 (2004)

    2004

  30. [38]

    K. Y. Bliokh and F. Nori, Relativistic Hall Effect, Phys. Rev. Lett.108, 120403 (2012)

    2012

  31. [39]

    S. W. Hancock, S. Zahedpour, A. Goffin, and H. M. Milchberg, Spatiotemporal Torquing of Light, Phys. Rev. X14, 011031 (2024)

    2024

  32. [40]

    K. Y. Bliokh, On the Ehrenfest theorem and centroids of relativistic particles, Phys. Lett. A542, 130425 (2025). 10 Appendix A: Centroids and OAM expressions in the nonparaxial regime The following tables show the expressions for the centroids and OAM calculated with Eqs. (6),...

    2025

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.