{"id":"7d0f3098-286a-4815-b3b4-c671a560562d","arxiv_id":"2510.20257","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"For aperture-limited Gaussian vortex beams, the dark core radius at the phase-imprinting plane is set by the first Bessel root of order ell, and the distance over which the core grows follows a D^{-3/2} scaling with fitted coefficient linear in ell.","lead":"The paper measures how a ring-shaped dark core appears and grows in laser beams that carry orbital angular momentum after they pass through small apertures, and gives simple scaling rules linking core size and growth distance to the beam's winding number and aperture size. These rules matter for building optical traps and atom interferometers that need a flat beam profile with a controlled phase twist.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"zV = c / D^(3/2) may be an artifact of the ad hoc interpolation Eq. 10; direct model/data extraction of zV is needed to confirm the scaling.","rationale":"The paper has two connected central claims. The vortex-core-radius relation (Eq. 39) is reasonably well grounded: Appendix D reduces the transcendental equation to the first root of J_ell, and the measured alpha(ell) trend is consistent with j_{ell,1}, albeit with a calibrated coefficient. That part does not need to be rejected. The uniformity-distance law (Eq. 12), however, is much less secure. It is not obtained by solving the propagation model; it is obtained by fitting a 7-parameter empirical surface (Eq. 10) to r0(z,D), solving for the crossing of a sqrt(2) threshold, and then fitting a power law. The exponent -3/2 is therefore a derived fitting parameter, not a predicted scaling. Since the interpolation weight W(D) is itself a power law with a fitted exponent, the resulting zV(D) could inherit a power-law form regardless of the underlying physics. The paper's own limitation statement says it is uncertain whether the linear trend between zV and ell is broadly applicable, and the numerical model's accuracy degrades for exactly the small-D regime that dominates the zV data. These are internal flags that the claim is conditional, not an established law. I agree with the reader's weakest-assumption analysis. The proposed test - extracting zV directly from the numerical model or raw data without passing through Eq. 10 - would settle whether D^(-3/2) is physical or an artifact. Until such a test is performed, keeping the verdict at CONDITIONAL is appropriate; the paper should not be rejected because Eq. 39 is plausible and the experiments are thoughtfully designed, but it should not be upgraded to an accepted law without independent verification.","tokens_in":15075,"tokens_out":9964,"duration_ms":85628,"concrete_test":"Run the authors' numerical PWE solver (or evaluate Eq. 3 directly) on a dense z grid for representative D values (0.2, 0.4, 0.6, 1, 2, 4 mm) and ell = 1, 2, 5, 10, with no use of Eq. 10. Locate zV directly as the smallest z at which r0(z,D) = sqrt(2) r0(0,D), interpolating between grid points, then fit log zV versus log D and c versus ell. If the -3/2 power and linear c(ell) survive, Eq. 12 is a genuine scaling; if the exponent drifts or c is nonlinear, Eq. 12 is an artifact of the interpolation. Ideally repeat on the raw experimental z-scans once data are available.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The second central law, zV = c / D^(3/2) with c linear in ell (Eq. 12), is not derived from the paraxial model; it is the result of a chain of fits. r0(z,D) is represented by Eq. 10, an ad hoc weighted average of the D=0.2 mm and no-LPF endpoint functions, with W(D) = (D/D0)^C7 and C7 < 0. zV for each D is then obtained by inverting Eq. 10 (r0(zV,D) = sqrt(2) r0(0,D)) and only afterwards fitted by a power law. Because zV inherits the assumed monotone power-law W(D), the observed D^(-3/2) dependence may be an artifact of the interpolation family rather than a physical scaling. No direct measurement of zV is reported - only five z positions per (ell,D) - and the D=5 mm point is excluded. The authors themselves note (Discussion) that it is uncertain whether the linear c(ell) trend is broadly applicable, and they acknowledge the hard-aperture numerical model deteriorates for small D, the regime where zV is largest. Thus the uniformity-distance claim is conditional at best. By contrast, r0 ~ j_{ell,1}/k_c (Eq. 39) is supported by the asymptotic derivation in Appendix D and the measured alpha(ell) trend; that part of the central claim is not the weak point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15513,"tokens_out":11241,"duration_ms":93890,"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":[{"comment":"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.","section":"Theory/Eq. (11); Appendix C"},{"comment":"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).","section":"Analysis, Eqs. (10)–(12)"},{"comment":"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.","section":"Results, Fig. 7"},{"comment":"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.","section":"Results, Fig. 5"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Eq. (12)"},{"comment":"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'.","section":"Fig. 7 caption"},{"comment":"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.","section":"Theory"},{"comment":"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.","section":"Fig. 4"}],"recommendation":"major_revision","confidential_remarks":"I see a clear path to revision: fix the factor-of-two convention in α/k_c and replace the Eq. (10)-derived zV analysis with direct extraction from the model or raw data. The experimental campaign is valuable and the Appendix D derivation is sound; I do not think rejection is warranted unless the reanalysis contradicts the claimed scalings."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper gives a clean, useful scaling for the vortex core radius at the phase-imprinting plane, and a much weaker empirical claim about the propagation distance zV. The first part holds up; the second is a fit to a fit and should not be treated as a law.\n\nWhat is new is genuinely useful to practitioners. They systematically measure r0 at d1 for ell=1,2,5,10 across LPF diameters from 0.2 to 5 mm and find r0 tracks alpha/j_l1 over k_c, Eq. 39. The Appendix D asymptotic derivation is clean: under large |beta| k_c^2, the derivative condition reduces to J_ell(alpha) ~ 0, so the first Bessel root appears naturally. The measured alpha(ell) = 1.03 ell + 3.02 is close to the Bessel trend j_l1 = 1.177 ell + 2.805, and the paper is honest that alpha is a calibration factor, not an independently predicted coefficient. The numerical model matches the data in the near field for large apertures; the authors themselves note it deteriorates for small D. They also correctly credit Vallone for the earlier observation that the intensity null is an aperture artifact. The citation pattern looks honest; the relationship to Vallone and to Janicijevic/Topuzoski is explicit.\n\nThe soft spot is zV. Every zV is extracted from Eq. 10, which is an ad hoc weighted interpolation between the D=0.2 mm and no-LPF curves, with weighting W(D)=(D/D0)^C7 and C7 fitted. So the ring-radius surfaces are fit first, zV is read off those fits, and the D^-3/2 exponent is then fit to those read-offs. That chain inherits the assumed monotone W(D); the power law is a summary of the interpolation family, not an independent physical scaling. Only five z positions per (ell,D) were used, the D=5 mm point is excluded, and no direct zV measurement is reported. The authors themselves flag uncertainty about the linear c(ell) trend. The stress-test note has this right. This does not damage the r0 result, but the two claims should be separated in revision.\n\nData and code are not public, which matters here because the zV exponent cannot be checked without the raw r0(D,z) data.\n\nWho this is for: experimentalists building aperture-limited Gaussian vortex systems—ring-trap atom interferometers, OAM communications, optical trapping. It is a practical characterization paper, not a theoretical advance, and it deserves a serious referee. I would accept and ask the referee to require a direct model/data extraction of zV, the D=5 mm data or an explanation, and ideally release of underlying data.","headline":"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.","tokens_in":15981,"tokens_out":3811,"would_cite":true,"duration_ms":33980,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["orbital angular momentum","optical vortex","Gaussian vortex beam","low-pass spatial filter","vortex core radius","Bessel function roots","uniformity distance","hypergeometric-Gaussian modes"],"falsifier":"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.","tokens_in":14964,"feed_emoji":"🌀","tokens_out":12357,"duration_ms":84604,"temperature":0.7,"pith_summary":"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.","feed_headline":"Vortex core size set by Bessel roots and aperture cutoff","feed_subtitle":"Core radius and √2-growth distance of OAM beams tied to low-pass filter size; useful for atom interferometry.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Vortex core radius scales with Bessel zeros and aperture cutoff","Aperture sets vortex core size: Bessel-root scaling law","Vortex growth distance scales as D^-3/2","OAM vortex core radius tied to Bessel zeros"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Vortex core radius scales with Bessel zeros and aperture cutoff","Aperture sets vortex core size: Bessel-root scaling law","Vortex growth distance scales as D^-3/2","OAM vortex core radius tied to Bessel zeros"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000945,"raw_usage":{"total_tokens":3866,"prompt_tokens":728,"completion_tokens":3138,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":3069}},"tokens_in":472,"tokens_out":3138,"duration_ms":19241,"temperature":1.0,"reasoning_tokens":3069,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:28:05.295312+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}