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REVIEW 4 major objections 5 minor 54 references

Vortex Propagation in Orbital Angular Momentum Beams and the Effects of a Limited Aperture

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper establishes that aperture-limited Gaussian vortex beams obey two scaling laws: r0 ≈ j_{ℓ,1}/k_c for the core radius and z_V = c/D^{3/2} for its growth distance.

desk verdict The r0-vs-aperture scaling is solid and useful; the zV power law is a fit to a fit and should be treated as preliminary. read the letter →

arxiv 2510.20257 v1 pith:BLBA4TQ4 submitted 2025-10-23 physics.optics

Ryan Husband , Jessica Eastman , Ryan J. Thomas , Simon A. Haine , Rhys H. Eagle , John D. Close , Samuel Legge This is my paper · ORCID
classification physics.optics
keywords orbitalangularmomentumopticalvortexGaussianbeamlow-passspatialfiltercoreradiusBesselfunctionrootsuniformitydistancehypergeometric-Gaussianmodes
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 claims that when a Gaussian beam is converted to a beam carrying orbital angular momentum by imprinting a helical phase, the vortex core that develops is not described by standard Laguerre–Gaussian propagation once the optical system has a finite aperture. Instead, the core radius at the phase-imprinting plane is set by the ratio of the first root of the Bessel function of order ℓ to the spatial cutoff frequency of the system (r0 ≈ j_{ℓ,1}/k_c), and the distance over which that core grows by √2 follows z_V = c / D^{3/2}, with c increasing linearly with ℓ. The paper verifies both laws experimentally for ℓ = 1, 2, 5, and 10 across a range of spatial low-pass filter diameters. A sympathetic reader would care because these two relations give a practical recipe for predicting and engineering vortex-core behavior in any phase-modulation OAM setup, including ring-trap atom interferometers.

What carries the argument

The argument is carried by two objects. The first is the dimensionless radial profile A(u) = ∫_0^u s^{ℓ+1} U((ℓ+2)/2, ℓ+1, β k_c^2 s^2/u^2) J_ℓ(s) ds, whose first maximum defines α = r0 k_c; an asymptotic expansion of the confluent hypergeometric U function reduces the stationarity condition to J_ℓ(α) ≈ 0, so α ≈ j_{ℓ,1}. The second is the empirical interpolation r0,ℓ(D,z) = [1−W(D)] r0,ℓ(0.2,z) + W(D) r0,ℓ(∞,z) with W(D)=(D/D0)^{C7}, which combines the small-aperture and no-aperture ring-radius behaviors; solving r0,ℓ(D,z_V)=√2 r0,ℓ(D,0) from this surface yields the z_V = c/D^{3/2} law. The hard-aperture transfer function H(k_r)=1 for k_r≤k_c, 0 otherwise, models the spatial LPF.

What would settle it

Measure r0(D,0) and z_V(D) for aperture diameters not used in the fits (e.g., D = 0.8, 1.2, 3 mm) at fixed ℓ, and check whether r0 k_c tracks j_{ℓ,1} and whether z_V D^{3/2} stays constant; a systematic drift in z_V D^{3/2} with D would falsify the power law, while a deviation of r0 k_c from j_{ℓ,1} would falsify Eq. 39.

Watch

Extended reading notes

Core claim

The central discovery is a pair of scaling relations for Gaussian vortex modes under aperture limitation. First, a vortex core forms whose radius at the phase-imprinting plane obeys r0 ≈ j_{ℓ,1}/k_c, where j_{ℓ,1} is the first positive zero of the ℓ-th Bessel function and k_c is the spatial cutoff; experimentally α = r0 k_c follows α(ℓ) = 1.03ℓ + 3.02, close to the Bessel-root trend j_{ℓ,1} = 1.177ℓ + 2.805. Second, defining z_V as the distance over which the primary ring radius grows by √2, the measured z_V follows z_V = c/D^{3/2}, with c = (3.50ℓ + 34.58)×10^{-2} in the experimental units. The numerical model agrees with data in the near-field regime, though its accuracy degrades for the s

Load-bearing premise

The load-bearing premise is that the fitted interpolation W(D)=(D/D0)^{C7} honestly captures how ring radius behaves at intermediate aperture sizes; if that weighting is not faithful, the D^{-3/2} law is an artifact of the fitting function rather than a physical scaling, and the hard-aperture model H(k_r) is only approximate for small apertures.

Editorial extensions

If this is right

  • For any OAM beam made by phase imprinting, the vortex core radius at the generation plane can be predicted from the aperture diameter, wavelength, and focal length via r0 ≈ j_{ℓ,1}/k_c, without full numerical propagation.
  • Smaller low-pass filters and larger ℓ both enlarge the vortex core at the phase-imprinting plane, so applications requiring a flat intensity profile must account for this trade-off.
  • The uniformity distance z_V can be tuned by choosing aperture size: a smaller aperture extends the distance over which the core size stays within √2 of its initial value, at the cost of a larger core and a radial intensity gradient.
  • The numerical model and the r0,ℓ(D,z) fit reproduce the measured ring radii in the near field for ℓ=1,2,5,10, supporting the use of these relations as design rules for GV-mode optical systems.
  • Because radial oscillations from high-frequency diffraction are suppressed by the low-pass filter, the filter both stabilizes the vortex core and introduces the radial gradient that motivated using Gaussian vortex beams in the first place.

Reading between the lines

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

  • If the D^{-3/2} law holds beyond the fitted range, it implies an explicit engineering trade-off: to double z_V for fixed ℓ one must shrink the aperture by a factor 2^{-2/3} ≈ 0.63, which in turn grows r0 by roughly 1/0.63 ≈ 1.59 via Eq. 39 — a quantitative version of the stability-versus-uniformity compromise the authors describe qualitatively.
  • The near-equality between the measured α(ℓ)=1.03ℓ+3.02 and the Bessel-root trend j_{ℓ,1}=1.177ℓ+2.805 suggests that the offset and slope of α(ℓ) may be universal up to circuit-dependent constants; a testable extension is to measure α(ℓ) in 4f systems with different focal lengths and check whether (α − a2)/a1 collapses onto j_{ℓ,1}-scaled curves.
  • The numerical staircase in r0 for small D suggests that tracking the maximum-intensity ring is not the most robust definition of core radius under strong filtering; an editorial extension would be to recompute z_V using the phase-singularity location or the ring's second moment, which might smooth the law and extend it to smaller apertures.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper studies the propagation of Gaussian vortex (GV) beams—Gaussian amplitude profiles with helical phase—through a finite aperture modeled as a low-pass spatial filter. A numerical paraxial model and a 4f imaging experiment with 12 hard apertures for ℓ = 1, 2, 5, 10 are presented. Two scaling laws are proposed: the vortex core radius at the phase-imprinting plane, r0 ≈ j_{ℓ,1}/k_c (Eq. 39), and the uniformity distance zV = c/D^{3/2} with c approximately linear in ℓ (Eq. 12), where zV is the distance over which the primary ring radius grows by √2. The first law is supported by an asymptotic derivation in Appendix D and by a measured linear α(ℓ) trend; the second is obtained by fitting an ad hoc interpolation surface (Eq. 10) and then fitting the resulting zV values.

Significance. If the advertised relations hold, the paper provides useful design rules for aperture-limited GV modes relevant to OAM applications and ring-trap atom interferometry. The experimental campaign is substantial, with four topological charges and a range of aperture diameters, and the Appendix D asymptotic derivation is clean and transparent. These strengths are real. However, the paper's headline uniformity-distance law is not independently derived and is entangled with the interpolation used to define it, and a factor-of-two inconsistency in the definition of α means the claimed support for Eq. 39 also needs reconciliation. With those points resolved, this would be a valuable contribution to the structured-light literature.

major comments (4)
  1. [Theory/Eq. (11); Appendix C] Eq. (11) defines α_{n,ℓ} = (2π/(λ f)) r_{0,ℓ}(D_n,0) D_n. Appendix C gives k_c = kD/(2f) = πD/(λ f), and Eq. (5) defines α = k_c r_0. Therefore the quantity in Eq. (11) equals 2k_c r_0, not α. Unless a different convention for k_c is intended, the measured α values in Fig. 6 are a factor of 2 larger than the α used in Eqs. (5) and (39). The apparent agreement with j_{ℓ,1} would then actually be agreement with 2j_{ℓ,1}, contradicting Eq. (39). Please state the convention explicitly and correct Eq. (6), Eq. (11), or Eq. (39) accordingly. This is central to the core-radius claim.
  2. [Analysis, Eqs. (10)–(12)] The zV(D) law is not a direct measurement. The surface r_{0,ℓ}(D,z) is built from the interpolation (10), whose weight W(D) = (D/D0)^{C7} is a power law with C7 < 0; zV is then read off from this fitted surface, and the D^{-3/2} exponent is itself a best-fit parameter. A power-law zV(D) is effectively inherited from the assumed power-law W(D), so Eq. (12) is a summary of the fit rather than an independent physical scaling. The exclusion of D = 5 mm from Fig. 7 further weakens the fit. I ask the authors to extract zV directly from the numerical model and, ideally, from raw r0(z) data at more than five z-positions per condition, without imposing Eq. (10), and to report those values and their residuals against Eq. (12).
  3. [Results, Fig. 7] Even if Eq. (12) is accepted as an empirical relation, the evidence for c ∝ ℓ rests on only four values (ℓ = 1, 2, 5, 10), and the paper itself states that it is uncertain whether the linear trend is broadly applicable. Please provide confidence intervals for the exponent p in zV = c/D^p, test the stability of c(ℓ) when D = 0.2 mm or D = 5 mm is removed, and specify the physical units in Eq. (12). Without this, the relation cannot be used as a predictive design rule.
  4. [Results, Fig. 5] The numerical model agreement is weakest exactly in the small-D regime where zV is largest: the authors state that model accuracy deteriorates with decreasing LPF size and that the data do not reproduce the predicted staircase behavior. Since the zV scaling is anchored at small D, the paper should quantify the model-data discrepancy (e.g., RMS error in r0 as a function of D) and show that the extracted zV values are not sensitive to the model's known inaccuracies in that regime. Currently the small-D comparison is only qualitative.
minor comments (5)
  1. [Throughout] Typographical issues: 'Seigman' should be 'Siegman'; 'Fourier plain' in Results should be 'plane'; the right-panel axis label of Fig. 7 appears garbled ('m5=2#10!2') and should be typeset properly.
  2. [Eq. (12)] The phrase 'arbitrary c ∈ R' is misleading because c must carry dimensions (zV has length, D has length, so c has length^{5/2}). Please nondimensionalize or state units explicitly.
  3. [Fig. 7 caption] The exclusion of the D = 5 mm data point is mentioned only in the main text; the caption should state this and give the quantitative threshold for 'below a measurable threshold'.
  4. [Theory] The claim that ℓ up to ~10^6 is within the paraxial regime for 1-mm-scale GV modes is asserted without a derivation or estimate; a brief justification or citation would help.
  5. [Fig. 4] The definition of r0 as the maximum of the orbitally integrated radial profile is implied but not stated explicitly. Please state how r0 was extracted from both data and model, and note any systematic differences.

Circularity Check

1 steps flagged · score 6.0 of 10

Partial circularity: Eq. 12 (zV = c/D^{3/2}) is a post-hoc fit summary of the Eq. 10 interpolation surface, not an independent prediction; Eq. 39 (r0 ≈ j_{ell,1}/k_c) is independently derived.

  1. fitted input called prediction [Analysis and Results/Discussion, Eqs. (9)–(12), Fig. 7]
    "Using Eq. 10, we measure zV as a function of D for each ℓ data set to find a simple relationship for diverging GV modes as shown in Fig. 7. ... For each D, zV was calculated as r0,ℓ(D,zV)=√2r0,ℓ(D,0), demonstrating the following power law coupling between D and zV independent of ℓ zV= c/D^{3/2} (12) ... Notably, the 3/2 power of D was a best fit parameter for each dataset independent of ℓ."

    zV is not independently predicted or directly measured as a scaling law. It is obtained by inverting the fitted surface r0,ℓ(z,D) of Eq. 10, whose D-dependence was constructed from the fitted power-law weighting W(D)=(D/D0)^C7 interpolating between the D=0.2 mm and no-LPF endpoints. The exponent -3/2 in Eq. 12 is then itself a best-fit parameter on those already-fit-derived zV values, and c(ℓ) is a linear fit of the fitted c values. Thus Eq. 12 is a summary of the same fitting family, inheriting its power-law form from Eq. 10 rather than testing it against an independent first-principles prediction. The paper's own caveat that it is uncertain whether the linear c(ℓ) trend is broadly applicable, and the exclusion of the D=5 mm point, underline that this part of the central claim reduces to

full rationale

The analytically derived core-radius relation r0 ≈ j_{ell,1}/k_c (Eq. 39, Appendix D) is not circular: it follows from the GV ansatz, the hard-aperture transfer function, an asymptotic expansion of the Tricomi function, and the resulting Bessel-root condition; the experimental α values are then compared to that prediction, not used to define it. The self-citation content is minimal and not load-bearing. The partial circularity lies in the second claimed law, Eq. 12: the zV values are read off from the fitted interpolation surface Eq. 10, and the D^{-3/2} exponent is itself a best-fit parameter on those same derived values. Consequently Eq. 12 is a compact description of the fitted surface rather than an independent physical prediction. This supports score 6: one central claimed relationship reduces by construction to its fitting inputs, while the other central relationship has genuine independent derivation and experimental support.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper introduces the defined quantity zV (uniformity distance), but this is a measurement definition rather than a postulated physical entity. No new forces, particles, or dimensions are introduced. The main burden is not an invented entity but the large number of fitted parameters: six empirical endpoint/interpolation functions and two scaling-law fits.

free parameters (6)
  • alpha(ell) calibration fit = 1.03 ell + 3.02
    Linear regression of measured alpha_n,ell in Fig. 6; used in Eq. 6 as a calibration factor, not predicted.
  • C1, C2, C3 per ell = separately fitted for ell = 1, 2, 5, 10
    Eq. 7 models ring radius at the smallest LPF (D = 0.2 mm) along z; an empirical endpoint function with no derivation.
  • C4, C5, C6 per ell = C6 ~ 0.5; other values fitted per ell
    Eq. 8 models the no-LPF ring radius along z as a power law; chosen by fit, not derived.
  • C7 and D0 in weighting W(D) = C7 < 0; D0 normalization length
    Eq. 9 interpolates between the small-aperture and no-aperture regimes using W(D) = (D/D0)^C7, fitted to the z = 0 data.
  • zV exponent p = p ~ 1.5
    In Eq. 12, zV = c/D^p, the 3/2 power is described as a best-fit parameter determined independently for each ell.
  • zV coefficient c = (3.50 ell + 34.58) x 10^-2 m^(5/2)
    Linear fit of c versus ell in Fig. 7; the paper explicitly treats c as an experimentally dependent calibration constant.
assumptions (6)
  • domain assumption The paraxial wave equation, Eq. 1, is valid for the beams and parameters studied.
    Invoked in the Theory section; the paper uses the upper bound ell ~ 10^6 from Refs. [40,41] to justify PWE for their beam sizes.
  • standard math Type I HyGG modes from the circular-beam basis correctly describe phase-imprinted Gaussian beams.
    Derived in Appendices A-B from the CiB basis; relies on cited mathematical results [26,27].
  • domain assumption Thin-element approximation: the GV mode G(r,q) e^{i ell theta} exists only in a narrow plane at z -> d1.
    Eq. 2 and Appendix B; necessary for using the GV field as the initial condition for propagation.
  • ad hoc to paper The hard low-pass transfer function H(kr), Eq. 31, models the physical aperture.
    The paper uses this transfer function in the analytic derivation and numerical model, while noting the model's accuracy deteriorates for small aperture diameters.
  • domain assumption Asymptotic expansion of the Tricomi U function requires |beta|^2 k_c^2 >> 1.
    Appendix D uses this expansion to reach the leading-order Bessel-root equation; the paper asserts it holds for most optical circuits.
  • ad hoc to paper Equation 10, a weighted interpolation between D = 0.2 mm and D = infinity, represents r0 across all D and z.
    The zV values are computed from this fitted surface; there is no physical derivation of the weighting function W(D).

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Cite this review

Pith. "Pith review of Vortex Propagation in Orbital Angular Momentum Beams and the Effects of a Limited Aperture." pith.science (2026). https://pith.science/paper/BLBA4TQ4

@misc{pith2026251020257,
  author       = {Pith},
  title        = {Pith review of: Vortex Propagation in Orbital Angular Momentum Beams and the Effects of a Limited Aperture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLBA4TQ4}},
  note         = {Machine review of arXiv:2510.20257}
}
abstract

When generating light with orbital angular momentum by imprinting orbital phase onto a standard Gaussian beam, it is often assumed that the propagation of the generated spatial mode is a Laguerre-Gaussian. However, the true propagation of this beam in a realistic, aperture-limited optical system is non-trivial and has not been thoroughly explored in existing literature. We explore a numerical model that shows the development of an optical vortex mode, propagating from the plane of phase modulation, and the relation of these dynamics to the orbital phase factor $\ell$ and the spatial bandwidth of the optical system. The results of this model are compared to experimental data for beams with $\ell$ values 1, 2, 5, and 10 propagating through a range of spatial filters, with the described model showing agreement in the near field regime.

Figures

Figures reproduced from arXiv: 2510.20257 by the authors.

Figure 1
Figure 1. Visual of a HyGG mode with ℓ = 1 forming from orbital phase modulation of a 780 nm wavelength Gaussian beam with a 1.52 mm beam waist, produced by solving the PWE numerically with a GV mode as an initial condition at 𝑑1. The primary ring radius, 𝑟0, is shown as a function of the optical axis, 𝑧. The Rayleigh range, 𝑧R, defines the propagation distance where the vortex core diameter stabilizes to that of half of the … view at source ↗
Figure 2
Figure 2. The top row depicts intensity images captured of GV modes across several [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Diagram of the setup as described in the text, where a HyGG [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (Left) A comparison between the data (integrated orbitally from an intensity [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Ring radii data sets and corresponding numerical model outputs for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: (Left) The vortex core size at the phase imprinting plane, as measured across [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: 0.2 0.6 1 2 3 4 D (mm) 0.2 1 5 10 20 50 zV (cm) Measuring zV of r0(z) ` =1 ` =2 ` =5 ` =10 1 2 5 10 ` 20 30 40 50 60 70 80 c (m5=2 # 10!2) Scaling Constant of zV for ` c = (3.50` + 34.58)#10!2 Data Fit to Data [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.