Beam quality measurement through off-axis optical vortex

Agnieszka Popio{\l}ek-Masajada; Jan Masajada; Mateusz Szatkowski

REVIEW 3 major objections 6 minor 16 references

Beam quality measurement through off-axis optical vortex

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the trajectory of an off-axis optical vortex, tracked with a Laguerre-Gaussian filter, exposes astigmatism as an inclination of the trajectory, and that the filter-based localization beats center-of-mass tracking.

desk verdict A modest but solid methods paper: LG-transform vortex localization is a genuine improvement over center-of-mass near the beam edge, though the accuracy claim lacks ground truth and the ω selection rule is underspecified. read the letter →

arxiv 1908.08879 v2 pith:T3GUBIFS submitted 2019-08-23 physics.optics

Mateusz Szatkowski , Agnieszka Popio{l}ek-Masajada , Jan Masajada This is my paper · ORCID

classification physics.optics

keywords beamqualitymeasurementopticalvortextrajectoryLaguerre-Gaussiantransformlocalizationastigmatismdetectionspatiallightmodulatorstructured

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 proposes that the trajectory traced by an optical vortex shifted off-axis is a workable, quantitative beam-quality probe, and that a new vortex-localization algorithm makes that probe objective. The method encodes the measured intensity with a Laguerre-Gaussian filter to produce a pseudo-complex signal, then defines the vortex point as the intersection of the zero isolines of its real and imaginary parts. The authors show this locates the vortex more reliably than the center-of-mass method, especially near the beam edge, where the two methods differed by 35 pixels in their example. Using the trajectories from x and y scans, they report that adding astigmatism tilts both trajectories, so trajectory shape can reveal a specific aberration without interferometric measurement. A sympathetic reader would care because it replaces visual inspection with an automatable criterion for beam quality and spatial light modulator correction.

What carries the argument

The central object is the Laguerre-Gaussian transform of the intensity image, a convolution whose Fourier-domain kernel is $\mathbf{LG}(f_x,f_y)=\rho \exp(-\rho^2/\omega^2)\exp(j\beta)$, with $\rho$ the radial spatial frequency and $\beta$ the azimuth. Its inverse Fourier transform has the form $(j\pi^2\omega^4)(x+jy)\exp(-\pi^2\omega^2(x^2+y^2))$, so the transform turns the real intensity $I(x,y)$ into a pseudo-complex signal $\tilde{I}(x,y)$. Near a vortex, the real and imaginary parts of $\tilde{I}$ are approximately planar, and their zero isolines $g_1=0$, $g_2=0$ intersect at candidate vortex points; the vortex point is the intersection with lowest intensity. The bandwidth $\omega$ sizes the filter to the vortex and is what makes one setting work across the dataset.

What would settle it

Simulate a Gaussian beam with a known vortex position and known astigmatism, run the Laguerre-Gaussian-transform localization over a grid of $\omega$ values and noise levels, and compare recovered vortex positions and trajectory angles to ground truth. If the localization error or the inferred trajectory inclination changes materially with $\omega$ or noise, the claimed objectivity and aberration readout fail; if the measured inclination does not scale with the added Zernike coefficient, the aberration claim fails.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the off-axis vortex trajectory, recovered by Laguerre-Gaussian-transform localization, acts as an aberration signature. In a corrected system the x- and y-scan trajectories are nearly straight and mutually perpendicular; after adding astigmatism with Zernike coefficient $Z_2^{-2} = 0.54\lambda$, both trajectories incline, with opposite senses tied to the sagittal and transverse foci. The localization algorithm behind this is claimed to be more accurate than center-of-mass tracking, with a 35-pixel discrepancy at the beam edge in the demonstrated case, and it can use fixed parameters across an entire dataset once the filter bandwidth is chosen.

Load-bearing premise

The method's accuracy rests on choosing the filter bandwidth $\omega$ correctly, but the paper gives no rule for how that choice is made, only saying the positions are accurate when $\omega$ is well determined.

Editorial extensions

If this is right

System astigmatism can be read directly from the inclination of vortex trajectories, giving a non-interferometric beam-quality check.
The localization algorithm can be automated for a full scan because the same parameters work for every image, unlike center-of-mass, which needs per-image thresholds.
The trajectory shapes can serve as an objective merit function for spatial light modulator correction, replacing visual inspection of the vortex image.
Vortex trajectory inspection near the beam edge is where the new method pays off, since center-of-mass fails there but the Laguerre-Gaussian transform still finds the vortex.
The same vortex-trajectory criterion can support exchanging a spiral phase plate for a spatial light modulator in scanning-microscope systems by verifying straight perpendicular trajectories.

Reading between the lines

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

The authors leave implicit that the trajectory inclination angle could be inverted to estimate the Zernike coefficient of astigmatism; a calibration with known coefficients would test that.
Because the method's stated objectivity depends on the unstated choice of $\omega$, a deterministic rule for selecting $\omega$ from the vortex ring size would be the natural next step; without it, automation still needs a human in the loop.
The same trajectory-shape logic might extend to other low-order aberrations such as coma or defocus, with each aberration predicted to deform x/y trajectories differently, but the paper only demonstrates astigmatism.
The 35-pixel advantage is a single demonstrated case; a broader error study over vortex positions and noise levels would establish whether the advantage is systematic.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a modest but usable methods paper. The new thing is using a Laguerre-Gaussian transform to localize the vortex core in off-axis vortex trajectory measurements, and it comes with code and experimental examples. It is not a breakthrough, but it is a reasonable improvement over center-of-mass for vortex localization near the beam edge, where the old method visibly fails. The paper is honest about its limitations.

What is genuinely good: the algorithm is clearly described, the MATLAB code is linked, and the experimental demonstration with astigmatism gives a qualitative proof of concept. The authors correctly attribute the LG transform to earlier speckle metrology work (Refs. 12–14) and their own prior trajectory work (Ref. 5). The trajectory inclination under astigmatism is a nice visual demonstration, and the writing is straightforward.

The soft spots are real but not fatal. The main accuracy claim—35 px improvement near the edge—rests on a single example with no ground truth. Center-of-mass is a biased estimator for a phase singularity, so beating it does not by itself establish correctness. The bandwidth ω in Eq. (2) is hand-tuned; the paper says positions are accurate “when bandwidth ω is well determined” but never says how to choose it or how sensitive results are to it. That matters for the claimed objectivity and automation. Also, the astigmatism link is qualitative: we see inclination but no quantitative relation between trajectory shape and the Zernike coefficient. That is a minor issue for a proof-of-concept, but it should be flagged.

If I were refereeing this, I would ask for a ground-truth test—simulated vortex images with known positions—and a sensitivity analysis over ω. The paper would be stronger with those. But the core idea is sound and the code makes it checkable. This is the kind of work that deserves a serious referee rather than a desk reject.

Referee Report

3 major / 6 minor

Summary. The paper proposes a beam-quality inspection method based on tracking an off-axis optical vortex generated by a spatial light modulator (SLM). When the vortex phase pattern is translated, the vortex point moves within the beam, and the shape of its trajectory is proposed as an indicator of optical aberrations. A new vortex localization algorithm is presented: the intensity image is transformed via a Laguerre-Gaussian filter to a pseudo-complex signal; the real and imaginary parts are locally approximated by planes (Eq. 4); their zero isolines are intersected (Eq. 6); and the final vortex point is selected as the intersection with the lowest intensity. The authors claim this method is more accurate than the previously used center-of-mass approach, especially near the beam edge, and report a 35-pixel difference in one example. Experiments with x- and y-scans, without and with a manually introduced astigmatism term (Z2^-2 = 0.54λ), show that astigmatism inclines the vortex trajectories. The MATLAB implementation is made available on GitHub.

Significance. If the localization accuracy claim is established, the proposed algorithm would be a practical, non-interferometric tool for vortex tracking in optical vortex scanning microscopy and for SLM-based beam-quality monitoring. The method is physically grounded in the known Laguerre-Gaussian transform and previous optical vortex metrology, and the code availability is a notable strength for reproducibility. The experimental demonstration of astigmatism-induced trajectory inclination is promising as a proof of principle. However, the current validation is qualitative and rests on an unstated calibration step for the filter bandwidth; the significance of the accuracy claim is therefore conditional on additional validation.

major comments (3)

[§3, Fig. 6] The central claim that the Laguerre-Gaussian localization method is more accurate than the center-of-mass method is not supported by the presented evidence, because there is no independent ground truth for the true vortex position in the experimental images. The center-of-mass estimator is biased for a phase singularity, so a 35-pixel disagreement in one image does not demonstrate that the new method is correct. I recommend adding a validation with synthetic images or simulated vortex beams with known singularity positions, or an independent interferometric phase measurement, and reporting localization errors as a function of vortex position, noise level, and the filter bandwidth ω.
[§3, Eq. (2)] The bandwidth ω in the Laguerre-Gaussian filter is the key free parameter of the algorithm, and the paper states that vortex positions are accurate only 'when bandwidth ω is well determined' without specifying how ω is selected or how sensitive the results are to it. Because the number of candidate intersections in Eq. (6) and the final chosen point depend on ω, the objectivity and accuracy claims are not yet established. A fixed, reproducible selection rule for ω and a sensitivity analysis over the full dataset are needed to support the method.
[§3, Fig. 7 and §4] The astigmatism demonstration is qualitative: the text reports that astigmatism led to inclination of both trajectories 'as was expected' but provides no quantitative model relating trajectory inclination to the Zernike coefficient, no repeatability data, and no estimate of how localization uncertainty propagates to the trajectory shape. Since the paper's title promises beam quality measurement rather than mere detection, the central claim would be strengthened by a testable relation between trajectory geometry and aberration coefficients, or at least an explicit error-bar analysis of the trajectories shown in Fig. 7.

minor comments (6)

[Abstract and §1] There are numerous typographical and grammatical errors, e.g., 'optic al', 'Problem of efficient beam evaluation is just as important', and 'none of correction methods can work'; the manuscript would benefit from a careful language edit.
[§2] The notation for the trajectory angle is inconsistent: the text uses α in one place and then writes 'the angle 𝑎 = 𝜋/2'; please unify the symbol.
[§3, Eq. (2)] In Eq. (2), ρ is defined as sqrt(x^2 + y^2) while β uses fx and fy; please clarify that x and y here denote spatial frequency coordinates, or change the notation to avoid confusion.
[§3, Eq. (4)] The region over which the least-squares plane approximation in Eq. (4) is fitted is not specified; please define the window size or neighborhood used, as this affects the reproducibility of the algorithm.
[References] References [7] and [11] are duplicates of the same paper, and references [13] and [14] also duplicate the same work; these should be consolidated.
[§3] The sentence 'The MATLAB code with the implemented vortex localization algorithm can be find in GitHub repository' contains a typo ('can be find'); also, please describe the center-of-mass method that the new algorithm is compared with, since this is not defined or cited in the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LG-transform vortex localization is implemented from externally established filter formalism, and the astigmatism trajectory demonstration is a qualitative experimental check rather than a derivation from fitted inputs.

full rationale

The paper's derivation chain is a measurement pipeline, not a first-principles prediction. The Laguerre-Gaussian transform used in Eqs. (1)-(3) is taken from prior external speckle-metrology work (refs. [8], [12]-[14]); the local plane fit and zero-isoline intersection in Eqs. (4)-(6) are standard numerical localization steps. The accuracy comparison with center-of-mass (Fig. 6) is an empirical claim, and while it lacks an independent ground truth - the 35 px difference does not by itself prove the LG result is correct, and the bandwidth omega in Eq. (2) is hand-selected ('when bandwidth omega is well determined') - this is an unvalidated calibration/correctness concern, not circularity: the algorithm's output is not defined in terms of the claimed trajectory outcome. The trajectory-inclination result for astigmatism (Fig. 7b) is a qualitative consistency check guided by a focal-plane picture (Fig. 8), not a quantity fitted to the observed trajectories. The authors' own prior work is cited for the OVSM trajectory framework ([5], [7], [10], [11]), but the present experiments are self-contained demonstrations and the LG-filter basis is external, so self-citation is not load-bearing. No equation reduces to its input by construction, and no fitted parameter is renamed as a prediction. The omega-selection ambiguity and lack of ground truth belong in correctness risk, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. It relies on a hand-tuned filter bandwidth as a free parameter and on several domain assumptions, most notably the heuristic lowest-intensity selection rule and the qualitative link between trajectory shape and aberration. These assumptions are not independently validated beyond the single astigmatism example.

free parameters (1)

LG filter bandwidth omega = not reported
Eq. (2) defines the LG filter with bandwidth omega; the text says positions are accurate 'when bandwidth omega is well determined' but no selection rule or sensitivity analysis is given. It is adjusted by hand to match the vortex size, and it affects the localized vortex positions.

assumptions (4)

domain assumption The LG transform of the intensity signal produces a pseudo-complex signal whose zero-isolines intersect at optical vortex points (Eqs. 1-3).
This follows the cited speckle-metrology literature (Refs. 8, 12-14) and is adopted here without independent derivation.
domain assumption The pseudo-complex signal near a vortex core is locally linear, so a least-squares plane fit is a valid approximation (Eq. 4).
The paper assumes that the real and imaginary parts of the transformed signal can be approximated by planes in the close vicinity of the vortex center, which is a standard but unverified local-smoothness assumption.
ad hoc to paper The true vortex point among candidate intersections is the one with the lowest intensity.
The paper selects the candidate vortex point with the lowest intensity as the final answer, but provides no justification or test of this heuristic, especially in noisy images.
domain assumption Off-axis vortex shifting reveals beam aberrations as changes in trajectory shape, e.g., astigmatism tilts the trajectories.
This is physically motivated and supported by the experiment, but the paper does not provide a quantitative derivation linking trajectory inclination to the astigmatism coefficient.

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Cite this review

Pith. "Pith review of Beam quality measurement through off-axis optical vortex." pith.science (2026). https://pith.science/paper/T3GUBIFS

@misc{pith2026190808879,
  author       = {Pith},
  title        = {Pith review of: Beam quality measurement through off-axis optical vortex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3GUBIFS}},
  note         = {Machine review of arXiv:1908.08879}
}

read the original abstract

One of the challenges for every optical system is preserving the quality of the used beam, which may be significantly reduced, due to the low condition of used optical elements or their misalignment. There are plenty of methods focused on the correction of the final beam, depending on the used entire optical system. The problem of efficient beam evaluation is just as important. So far, most of the measurements, are based on visual inspection, which is not always enough, especially when the high quality of the beam is required. Novel approaches use structured light to increase beam sensitivity for any imperfections. In this paper, we present an approach, which uses optical vortex shifted off-axis for a beam quality measurement. It uses SLM as a vortex generating element, which is shifted off-axis by proper hologram transformation. Tracking of the vortex trajectory may provide information about beam quality and aberration of an optical system. The new vortex localization algorithm will be presented.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Strehl ratio for primary aberrations in terms of their aberration variance,

Mahajan, V. N., “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73(6), 860 (1983)

work page 1983

[2] [2]

Roadmap on superoscillations,

Berry, M., Zheludev, N., Aharonov, Y., Colombo, F., Sabadini, I., Struppa, D. C., Tollaksen, J., Rogers , E. T. F., Qin, F., Hong, M., Luo, X., Remez, R., Arie, A., Götte, J. B., Dennis, M. R., Wong, A. M. H., Eleftheriades, G. V, Eliezer, Y., Bahabad, A., et al., “Roadmap on superoscillations,” J. Opt. 21(5), 053002 (2019)

work page 2019

[3] [3]

Wavefront correction of spatial light modulators using an optical vortex image,

Jesacher, A., Schwaighofer, A., Fürhapter, S., Maurer, C., Bernet, S. and Ritsch-Marte, M., “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15(9), 5801 (2007)

work page 2007

[4] [4]

Aberration correction in holographic optical tweezers using a high-order optical vortex,

Liang, Y., Cai, Y., Wang, Z., Lei, M., Cao, Z., Wang, Y., Li, M., Yan, S., Bianco, P. R. and Yao, B., “Aberration correction in holographic optical tweezers using a high-order optical vortex,” Appl. Opt. 57(13), 3618–3623 (2018)

work page 2018

[5] [5]

Optical vortex trajectory as a merit function for spatial light modulator correction,

Szatkowski, M., Popiołek Masajada, A. and Masajada, J., “Optical vortex trajectory as a merit function for spatial light modulator correction,” Opt. Lasers Eng. 118(October 2018), 1–6 (2019)

work page 2019

[6] [6]

Optical vortex scanning inside th e Gaussian beam,

Masajada, J., Leniec, M. and Augustyniak, I., “Optical vortex scanning inside th e Gaussian beam,” J. Opt. 13(3) (2011)

work page 2011

[7] [7]

Internal scanning method as unique imaging method of optical vortex scanning microscope,

Popiołek-Masajada, A., Masajada, J. and Szatkowski, M., “Internal scanning method as unique imaging method of optical vortex scanning microscope,” Opt. Lasers Eng. 105(November 2017), 201–208 (2018)

work page 2018

[8] [8]

Optical vortex metrology for nanometric speckle displacement measurement,

Wang, W., Yokozeki, T., Ishijima, R., Wada, A., Miyamoto, Y., Takeda, M. and Hanson, S. G., “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14(1), 120 (2006)

work page 2006