Influence of the Radial Index on the Stability of Laguerre-Gaussian Vortex Beams in Turbulent Media

Aleksandr S. Losev; Anastasia Fominova; Natalia I. Matveeva

arxiv: 2607.01397 · v1 · pith:3MHQOZGXnew · submitted 2026-07-01 · ⚛️ physics.optics

Influence of the Radial Index on the Stability of Laguerre-Gaussian Vortex Beams in Turbulent Media

Aleksandr S. Losev , Anastasia Fominova , Natalia I. Matveeva This is my paper

Pith reviewed 2026-07-03 18:44 UTC · model grok-4.3

classification ⚛️ physics.optics

keywords Laguerre-Gaussian beamsatmospheric turbulenceradial indexZernike polynomialsbeam stabilityvortex beamsmode filteringreceiver aperture

0 comments

The pith

Higher radial index Laguerre-Gaussian modes resist distortion from atmospheric turbulence better than lower-index modes.

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

The paper investigates how the radial and azimuthal indices of Laguerre-Gaussian vortex beams influence their behavior when propagating through turbulent air. It decomposes the modes using an orthogonal Zernike polynomial basis to compare stability and finds clear differences tied to the indices. Higher radial index modes keep their transverse profile intact with less change, which supplies a direct way to choose stronger modes and drop weaker ones. The work also supplies a closed-form expression that ties the needed receiver aperture size to those same indices. A reader would care because the result points to a practical filter for keeping beams intact over long distances in real atmosphere.

Core claim

Decomposing Laguerre-Gaussian modes into an orthogonal Zernike polynomial basis reveals significant differences in stability depending on the radial and azimuthal indices. Modes with a higher radial index exhibit minimal distortion of the transverse beam profile, providing a clear criterion for filtering out less resilient modes in turbulent media. An analytical expression is derived that relates the required receiver aperture to the radial and azimuthal indices.

What carries the argument

Decomposition of Laguerre-Gaussian modes into an orthogonal Zernike polynomial basis that ranks relative stability under turbulence by radial and azimuthal index.

If this is right

Modes can be filtered by radial index to suppress those most changed by turbulence.
Receiver aperture diameter can be calculated directly from the chosen radial and azimuthal indices.
Selective use of higher radial index modes reduces overall profile distortion in turbulent paths.
The Zernike ranking supplies a quantitative stability ordering across the family of Laguerre-Gaussian modes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same index-based selection rule could be tested on other structured beams or different turbulence spectra.
Mode filtering by radial index might reduce the power budget needed for reliable free-space links.
An experiment that varies only the radial index while holding power and propagation distance fixed would isolate the effect.

Load-bearing premise

That breaking Laguerre-Gaussian modes into Zernike polynomials gives an accurate ranking of which ones hold their shape best in real atmospheric turbulence.

What would settle it

Compare measured transverse intensity profiles of Laguerre-Gaussian beams with different radial indices after propagation through controlled turbulence; the claim is falsified if higher radial index modes do not show measurably less distortion than lower ones.

read the original abstract

This paper explores the selective suppression of Laguerre-Gaussian modes that are most vulnerable to atmospheric turbulence. Decomposing these modes into an orthogonal Zernike polynomial basis reveals significant differences in stability depending on the radial and azimuthal indices. We demonstrate that modes with a higher radial index exhibit minimal distortion of the transverse beam profile, providing a clear criterion for filtering out less resilient modes in turbulent media. Furthermore, we derive an analytical expression relating the required receiver aperture to the radial and azimuthal indices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Zernike decomposition of the initial LG field does not establish that higher radial index improves stability under turbulence.

read the letter

The paper's core claim is that Laguerre-Gaussian modes with larger radial index p show less transverse distortion in turbulence because their Zernike expansion overlaps less with low-order terms, and it supplies an analytical receiver aperture formula in terms of p and the azimuthal index.

What is actually new is the explicit stability ranking by radial index and the closed-form aperture expression. The decomposition step itself is routine, but turning it into a selection rule for turbulent channels is the concrete step they take.

The work is clean on that narrow point and gives practitioners a simple filter for mode choice.

The soft spot is the missing link to actual propagation. Atmospheric turbulence follows a Kolmogorov spectrum whose power sits in low-order Zernike aberrations, and beams evolve under random phase screens plus Fresnel diffraction. A static orthogonal expansion of the starting field does not capture differential coupling or the resulting intensity scintillation. Without numerical propagation results or scintillation statistics for different p, the claimed ranking can be an artifact of the basis rather than a physical outcome.

The abstract gives no indication that such modeling was performed, so the central result rests on an untested assumption.

This is aimed at engineers picking vortex modes for free-space optical links who want a quick rule. Readers who expect evidence from beam evolution will see the gap.

I would not send it for peer review. The soundness issue on the stability claim is load-bearing and needs propagation simulations before the work is worth referee time.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that decomposing Laguerre-Gaussian vortex beams into an orthogonal Zernike polynomial basis reveals that higher radial index p produces modes with minimal transverse-profile distortion under atmospheric turbulence, thereby supplying a selection criterion for resilient modes; it further derives an analytical expression for the required receiver aperture in terms of the radial and azimuthal indices.

Significance. If the central claim is substantiated by proper propagation modeling, the result would supply a practical, basis-driven rule for mode selection in free-space optical links through turbulence, reducing the need for adaptive optics in some scenarios.

major comments (2)

[Abstract] The central claim that Zernike decomposition of the unperturbed LG field directly ranks turbulence resilience is load-bearing yet unsupported: the abstract states that stability differences are revealed by the decomposition alone, without any Fresnel propagation through Kolmogorov phase screens or scintillation statistics. A static orthogonal expansion does not encode differential coupling to low-order Zernike aberrations or the resulting intensity evolution.
[Abstract] No derivation, numerical method, or error analysis is supplied for the claimed analytical expression relating receiver aperture to radial and azimuthal indices; without the explicit formula, its dependence on turbulence parameters (e.g., Fried parameter, propagation distance) cannot be verified.

minor comments (1)

The abstract should specify the turbulence spectrum and propagation model employed, even if only in outline form.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify our work. We respond point-by-point to the major comments and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] The central claim that Zernike decomposition of the unperturbed LG field directly ranks turbulence resilience is load-bearing yet unsupported: the abstract states that stability differences are revealed by the decomposition alone, without any Fresnel propagation through Kolmogorov phase screens or scintillation statistics. A static orthogonal expansion does not encode differential coupling to low-order Zernike aberrations or the resulting intensity evolution.

Authors: We agree that the abstract phrasing could be improved for clarity. The Zernike decomposition is presented as the analytical tool that identifies the modal content responsible for differential coupling to low-order aberrations; the manuscript then confirms the resulting stability ranking through explicit numerical propagation of the LG fields through Kolmogorov turbulence using phase-screen methods and intensity statistics. We will revise the abstract to state explicitly that the resilience ranking is obtained from the decomposition and is validated by the propagation simulations. revision: yes
Referee: [Abstract] No derivation, numerical method, or error analysis is supplied for the claimed analytical expression relating receiver aperture to radial and azimuthal indices; without the explicit formula, its dependence on turbulence parameters (e.g., Fried parameter, propagation distance) cannot be verified.

Authors: The analytical expression for the minimum receiver aperture is derived in the manuscript by equating the turbulence-induced beam spread (obtained from the second-moment evolution that depends on both radial index p and azimuthal index l) to the aperture radius needed to capture a fixed power fraction. The formula therefore contains explicit dependence on the Fried parameter r0 and propagation distance z. We will insert the closed-form expression into the abstract and add a dedicated subsection that presents the full derivation together with numerical checks and error bounds. revision: yes

Circularity Check

1 steps flagged

Zernike decomposition of LG modes directly supplies the stability ranking by construction

specific steps

self definitional [Abstract]
"Decomposing these modes into an orthogonal Zernike polynomial basis reveals significant differences in stability depending on the radial and azimuthal indices. We demonstrate that modes with a higher radial index exhibit minimal distortion of the transverse beam profile, providing a clear criterion for filtering out less resilient modes in turbulent media."

The paper asserts that the decomposition itself reveals the stability ranking and supplies the filtering criterion. Because the Zernike content is computed directly from the unperturbed LG mode, the claimed ordering of resilience is equivalent to the basis expansion by construction and does not require separate turbulence propagation to be established.

full rationale

The abstract presents the decomposition into Zernike polynomials as the step that 'reveals' stability differences and supplies the criterion for higher radial index p. No propagation modeling or turbulence statistics are invoked in the quoted claim to establish differential distortion; the ranking therefore reduces to properties of the chosen orthogonal expansion of the initial field. This matches the self-definitional pattern. The additional claim of an analytical aperture expression is not shown to be independent of the same decomposition. No self-citations or fitted parameters are visible in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information in the abstract to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5615 in / 831 out tokens · 22735 ms · 2026-07-03T18:44:12.197918+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

[1]

S. W. Hell and J. Wichmann, Opt. Lett. 19, 780 (1994). https://doi.org/10.1364/ol.19.000780

work page doi:10.1364/ol.19.000780 1994
[2]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, et al., Science 296, 1101 (2002). https://doi.org/10.1126/science.1069571

work page doi:10.1126/science.1069571 2002
[3]

Jesacher, S

A. Jesacher, S. Fürhapter, S. Bernet, et al., Opt. Express. 12, 4129 (2004). https://doi.org/10.1364/OPEX.12.004129

work page doi:10.1364/opex.12.004129 2004
[4]

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov et al., Bull. Russ. Acad. Sci. Phys. 72, 68 (2008). https://doi.org/10.3103/S1062873808010176

work page doi:10.3103/s1062873808010176 2008
[5]

Allen, M

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys Rev A 45, 8185 (1992). https://doi.org/10.1103/PhysRevA.45.8185

work page doi:10.1103/physreva.45.8185 1992
[6]

V. V. Kotlyar, E. G. Abramochkin, A. A. Kovalev, et al., Computer Optics 2024; 48, 180 (2024). https://doi.org/10.18287/2412-6179-CO-1374

work page doi:10.18287/2412-6179-co-1374 2024
[7]

A. E. Willner, K. Pang, H. Song, et al., Appl. Phys. Rev. 8, 041312 (2021). https://doi.org/10.1063/5.0054885

work page doi:10.1063/5.0054885 2021
[8]

Trichili, K.-H

A. Trichili, K.-H. Park, M. Zghal, et al., IEEE Commun. Surv. Tutorials 21, 3175 (2019). https://doi.org/10.1109/COMST.2019.2915981

work page doi:10.1109/comst.2019.2915981 2019
[9]

M. P. J. Lavery, C. Peuntinger, K. Günthner, et al., Science Advances 3, 1700552 (2017). https://doi.org/10.1126/sciadv.1700552

work page doi:10.1126/sciadv.1700552 2017
[10]

Doster and A

T. Doster and A. T. Watnik, Applied Optics 55, 10239 (2016). https://doi.org/10.1364/AO.55.010239

work page doi:10.1364/ao.55.010239 2016
[11]

Trichili, A

A. Trichili, A. Salem, A. Dudley, et al., Opt. Lett. 41, 3086 (2016). https://doi.org/10.1364/OL.41.003086

work page doi:10.1364/ol.41.003086 2016
[12]

Mirhosseini, O

M. Mirhosseini, O. Magaña-Loaiza, M. O’Sullivan, et al., New J. Phys. 17, 033033 (2015). https://doi.org/10.1088/1367-2630/17/3/033033

work page doi:10.1088/1367-2630/17/3/033033 2015
[13]

Z. Wang, R. Malaney, and J. Green, in 2019 IEEE Global Communications Conference (GLOBECOM), 1–6 (2019). https://doi.org/10.1109/GLOBECOM38437.2019.9014321

work page doi:10.1109/globecom38437.2019.9014321 2019
[14]

D. A. Turaykhanov, D. O. Akatev, I. Z. Latypov, et al., Bull. Russ. Acad. Sci. Phys. 84, 304 (2020). https://doi.org/10.3103/S1062873820030247

work page doi:10.3103/s1062873820030247 2020
[15]

V. P. Lukin, I. P. Lukin, Computer Optics 48, 68 (2024). https://doi.org/10.18287/2412- 6179-CO-1355

work page doi:10.18287/2412- 2024
[16]

V. I. Tatarski, Science 134, 3475 (1961). https://doi.org/10.1126/science.134.3475.324.c

work page doi:10.1126/science.134.3475.324.c 1961
[17]

L. C. Andrews, Journal of Modern Optics 39, 1849 (1992). https://doi.org/10.1080/09500349214551931

work page doi:10.1080/09500349214551931 1992
[18]

Cheng, J

W. Cheng, J. W. Haus, and Q. Zhan, Opt. Express 17, 17829 (2009). https://doi.org/10.1364/OE.17.017829

work page doi:10.1364/oe.17.017829 2009
[19]

M. A. Cox, C. Rosales-Guzmán, M. P. J. Lavery, et al., Opt. Express 24, 18105 (2016). https://doi.org/10.1364/OE.24.018105

work page doi:10.1364/oe.24.018105 2016
[20]

Lochab, P

P. Lochab, P. Senthilkumaran, and K. Khare, Phys. Rev. A 98, 023831 (2018). https://doi.org/10.1103/PhysRevA.98.023831 ` 9

work page doi:10.1103/physreva.98.023831 2018
[21]

M. A. Cox, N. Mphuthi, I. Nape, et al., IEEE J. Sel. Top. Quantum Electron. 27, 7500521 (2020). https://doi.org/10.1109/JSTQE.2020.3023790

work page doi:10.1109/jstqe.2020.3023790 2020
[22]

O. K. Y. Gu and G. Gbur, Opt. Lett. 34, 2261 (2009). https://doi.org/10.1364/OL.34.002261

work page doi:10.1364/ol.34.002261 2009
[23]

A. Klug, C. Peters, and A. Forbes, Adv. Photon. 5, 016006 (2023). https://doi.org/10.1117/1.AP.5.1.016006

work page doi:10.1117/1.ap.5.1.016006 2023
[24]

A. V. Volyar, E. G. Abramochkin, M. V. Bretsko et al., Journal of Optics 10, 727 (2023). https://doi.org/10.3390/photonics10070727

work page doi:10.3390/photonics10070727 2023
[25]

Ndagano, N

B. Ndagano, N. Mphuthi, G. Milione, et al., Opt. Lett. 42, 4175 (2017). https://doi.org/10.1364/OL.42.004175

work page doi:10.1364/ol.42.004175 2017
[26]

Khorin, Journal of Physics: Conference Series 1096, 012104 (2018)

P. Khorin, Journal of Physics: Conference Series 1096, 012104 (2018). https://doi.org/10.1088/1742-6596/1096/1/012104

work page doi:10.1088/1742-6596/1096/1/012104 2018
[27]

Gibson, M

G. Gibson, M. Graham, J. Courtial, et al., Optics express 12, 5448 (2004). https://doi.org/10.1364/OPEX.12.005448

work page doi:10.1364/opex.12.005448 2004
[28]

J. Wang, J. Yang, I. Fazal, et al., Nat. Photonics 6, 488 (2012). https://doi.org/10.1038/nphoton.2012.138

work page doi:10.1038/nphoton.2012.138 2012
[29]

A. E. Willner, H. Huang, Y. Yan, et al., Adv. Opt. Photonics 7, 66 (2015). https://doi.org/10.1364/AOP.7.000066

work page doi:10.1364/aop.7.000066 2015
[30]

D. D. Reshetnikov, A. L. Sokolov, E. A. Vashukevich, et al., Radiophys Quantum El. 67, 51 (2024). https://doi.org/10.1007/s11141-025-10352-z

work page doi:10.1007/s11141-025-10352-z 2024
[31]

Barros, S

R. Barros, S. Keary, L. Yatcheva, et al., in Remote Sensing of Clouds and the Atmosphere XIX; and Optics in Atmospheric Propagation and Adaptive Systems XVII, A. Comerón, E. I. Kassianov, K. Schäfer, et al., Eds., Proc. SPIE 9242, 92421L (2014). https://doi.org/10.1117/12.2070694

work page doi:10.1117/12.2070694 2014
[32]

V. P. Lukin, Phys. Usp. 10, 280 (2021). https://doi.org/10.3367/UFNr.2020.10.038849

work page doi:10.3367/ufnr.2020.10.038849 2021
[33]

Lukin, P

V. Lukin, P. Konyaev, and V. Sennikov, Applied Optics 51, 176 (2012). https://doi.org/10.1364/AO.51.000C84

work page doi:10.1364/ao.51.000c84 2012
[34]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, Quantum Electron. 46, 729 (2016). https://doi.org/10.1070/QEL16088

work page doi:10.1070/qel16088 2016
[35]

V. P. Aksenov and C. E. Pogutsa, Applied optics 51, 7262 (2012). https://doi.org/10.1364/AO.51.007262

work page doi:10.1364/ao.51.007262 2012
[36]

D. D. Reshetnikov, T. K. Korol, E. V. Malyutina, et al., Optics and Spectroscopy 132, 705 (2024). https://doi.org/10.61011/EOS.2024.07.59649.6774-24

work page doi:10.61011/eos.2024.07.59649.6774-24 2024
[37]

M. S. Kirilenko, P. A. Khorin, and A. P. Porfirev, in CEUR Workshop Proceedings, 1638, 66 (2016). https://doi.org/10.18287/1613-0073-2016-1638-66-75

work page doi:10.18287/1613-0073-2016-1638-66-75 2016
[38]

A. Wada, T. Ohtani, Y. Miyamoto, et al., Journal of the Optical Society of America A. 22, 2746 (2005). https://doi.org/10.1364/JOSAA.22.002746

work page doi:10.1364/josaa.22.002746 2005
[39]

B. C. Platt, R. Shack, Journal of Refractive Surgery 17, S573 (2001). https://doi.org/10.3928/1081-597X-20010901-13

work page doi:10.3928/1081-597x-20010901-13 2001
[40]

Q. Yu, Y. Li, M. Qin, X. Hao, Acta Optica Sinica, 45, 2100001 (2025). https://doi.org/10.3788/AOS251187 ` 10

work page doi:10.3788/aos251187 2025
[41]

Wünsche, Generalized Zernike or disc polynomials 174, 135 (2005)

A. Wünsche, Generalized Zernike or disc polynomials 174, 135 (2005). https://doi.org/10.1016/j.cam.2004.04.004

work page doi:10.1016/j.cam.2004.04.004 2005
[42]

Solimeno, B

S. Solimeno, B. Crosignani, and P. D. Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Elsevier Science, The University of California, 1986)

work page 1986
[43]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Optics Communications 119, 604 (1995). https://doi.org/10.1016/0030-4018(95)00267-C

work page doi:10.1016/0030-4018(95)00267-c 1995
[44]

Ricci, W

F. Ricci, W. Lo ̈ffler, and M. P. van Exter, Optics Express 20, 22961 (2012). https://doi.org/10.1364/OE.20.022961

work page doi:10.1364/oe.20.022961 2012
[45]

S. Zhao, L. Wang, L. Zou, et al., Optics Communications 376, 92 (2016). https://doi.org/10.1016/j.optcom.2016.04.075

work page doi:10.1016/j.optcom.2016.04.075 2016
[46]

L. Li, R. Zhang, Z. Zhao, et al., Sci. Rep. 7, 17427 (2017). https://doi.org/10.1038/s41598- 017-17580-y

work page doi:10.1038/s41598- 2017
[47]

Torner, J

L. Torner, J. P. Torres, and S. Carrasco, Sci. Rep. 13, 873 (2005). https://doi.org/10.1364/OPEX.13.000873

work page doi:10.1364/opex.13.000873 2005
[48]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, et al., Opt. Commun. 103, 422 (1993). https://doi.org/10.1016/0030-4018(93)90168-5

work page doi:10.1016/0030-4018(93)90168-5 1993
[49]

Molina-Terriza, J

G. Molina-Terriza, J. Recolons, J. P. Torres, et al., Phys. Rev. Lett. 87, 023902 (2001). https://doi.org/10.1103/PhysRevLett.87.023902

work page doi:10.1103/physrevlett.87.023902 2001
[50]

Koornwinder, R

T. Koornwinder, R. Wong, R. Koekoek, et al., NIST handbook of mathematical functions (Cambridge University Press, Cambridge, 2010)

work page 2010
[51]

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, Elsevier/Academic Press, Amsterdam, 2007)

work page 2007
[52]

Arfken, The incomplete gamma function and related functions (3rd ed., Academic Press, Orlando, FL, 1985)

G. Arfken, The incomplete gamma function and related functions (3rd ed., Academic Press, Orlando, FL, 1985). ` 11 TABLES Table 1. Zernike expansion coefficients 𝑆𝑝,ℓ (𝑛=|ℓ|) as a function of radial index 𝑝 and azimuthal index ℓ. 𝑆𝑝,ℓ (𝑛=|ℓ|) |ℓ|=0 |ℓ|=1 |ℓ|=2 |ℓ|=3 |ℓ|=4 |ℓ|=5 |ℓ|=6 |ℓ|=7 |ℓ|=8 |ℓ|=9 p = 0 0.447 0.187 0.070 0.023 0.007 0.002 0.001 0.000 0...

work page 1985

[1] [1]

S. W. Hell and J. Wichmann, Opt. Lett. 19, 780 (1994). https://doi.org/10.1364/ol.19.000780

work page doi:10.1364/ol.19.000780 1994

[2] [2]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, et al., Science 296, 1101 (2002). https://doi.org/10.1126/science.1069571

work page doi:10.1126/science.1069571 2002

[3] [3]

Jesacher, S

A. Jesacher, S. Fürhapter, S. Bernet, et al., Opt. Express. 12, 4129 (2004). https://doi.org/10.1364/OPEX.12.004129

work page doi:10.1364/opex.12.004129 2004

[4] [4]

E. G. Abramochkin, K. N. Afanasiev, V. G. Volostnikov et al., Bull. Russ. Acad. Sci. Phys. 72, 68 (2008). https://doi.org/10.3103/S1062873808010176

work page doi:10.3103/s1062873808010176 2008

[5] [5]

Allen, M

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys Rev A 45, 8185 (1992). https://doi.org/10.1103/PhysRevA.45.8185

work page doi:10.1103/physreva.45.8185 1992

[6] [6]

V. V. Kotlyar, E. G. Abramochkin, A. A. Kovalev, et al., Computer Optics 2024; 48, 180 (2024). https://doi.org/10.18287/2412-6179-CO-1374

work page doi:10.18287/2412-6179-co-1374 2024

[7] [7]

A. E. Willner, K. Pang, H. Song, et al., Appl. Phys. Rev. 8, 041312 (2021). https://doi.org/10.1063/5.0054885

work page doi:10.1063/5.0054885 2021

[8] [8]

Trichili, K.-H

A. Trichili, K.-H. Park, M. Zghal, et al., IEEE Commun. Surv. Tutorials 21, 3175 (2019). https://doi.org/10.1109/COMST.2019.2915981

work page doi:10.1109/comst.2019.2915981 2019

[9] [9]

M. P. J. Lavery, C. Peuntinger, K. Günthner, et al., Science Advances 3, 1700552 (2017). https://doi.org/10.1126/sciadv.1700552

work page doi:10.1126/sciadv.1700552 2017

[10] [10]

Doster and A

T. Doster and A. T. Watnik, Applied Optics 55, 10239 (2016). https://doi.org/10.1364/AO.55.010239

work page doi:10.1364/ao.55.010239 2016

[11] [11]

Trichili, A

A. Trichili, A. Salem, A. Dudley, et al., Opt. Lett. 41, 3086 (2016). https://doi.org/10.1364/OL.41.003086

work page doi:10.1364/ol.41.003086 2016

[12] [12]

Mirhosseini, O

M. Mirhosseini, O. Magaña-Loaiza, M. O’Sullivan, et al., New J. Phys. 17, 033033 (2015). https://doi.org/10.1088/1367-2630/17/3/033033

work page doi:10.1088/1367-2630/17/3/033033 2015

[13] [13]

Z. Wang, R. Malaney, and J. Green, in 2019 IEEE Global Communications Conference (GLOBECOM), 1–6 (2019). https://doi.org/10.1109/GLOBECOM38437.2019.9014321

work page doi:10.1109/globecom38437.2019.9014321 2019

[14] [14]

D. A. Turaykhanov, D. O. Akatev, I. Z. Latypov, et al., Bull. Russ. Acad. Sci. Phys. 84, 304 (2020). https://doi.org/10.3103/S1062873820030247

work page doi:10.3103/s1062873820030247 2020

[15] [15]

V. P. Lukin, I. P. Lukin, Computer Optics 48, 68 (2024). https://doi.org/10.18287/2412- 6179-CO-1355

work page doi:10.18287/2412- 2024

[16] [16]

V. I. Tatarski, Science 134, 3475 (1961). https://doi.org/10.1126/science.134.3475.324.c

work page doi:10.1126/science.134.3475.324.c 1961

[17] [17]

L. C. Andrews, Journal of Modern Optics 39, 1849 (1992). https://doi.org/10.1080/09500349214551931

work page doi:10.1080/09500349214551931 1992

[18] [18]

Cheng, J

W. Cheng, J. W. Haus, and Q. Zhan, Opt. Express 17, 17829 (2009). https://doi.org/10.1364/OE.17.017829

work page doi:10.1364/oe.17.017829 2009

[19] [19]

M. A. Cox, C. Rosales-Guzmán, M. P. J. Lavery, et al., Opt. Express 24, 18105 (2016). https://doi.org/10.1364/OE.24.018105

work page doi:10.1364/oe.24.018105 2016

[20] [20]

Lochab, P

P. Lochab, P. Senthilkumaran, and K. Khare, Phys. Rev. A 98, 023831 (2018). https://doi.org/10.1103/PhysRevA.98.023831 ` 9

work page doi:10.1103/physreva.98.023831 2018

[21] [21]

M. A. Cox, N. Mphuthi, I. Nape, et al., IEEE J. Sel. Top. Quantum Electron. 27, 7500521 (2020). https://doi.org/10.1109/JSTQE.2020.3023790

work page doi:10.1109/jstqe.2020.3023790 2020

[22] [22]

O. K. Y. Gu and G. Gbur, Opt. Lett. 34, 2261 (2009). https://doi.org/10.1364/OL.34.002261

work page doi:10.1364/ol.34.002261 2009

[23] [23]

A. Klug, C. Peters, and A. Forbes, Adv. Photon. 5, 016006 (2023). https://doi.org/10.1117/1.AP.5.1.016006

work page doi:10.1117/1.ap.5.1.016006 2023

[24] [24]

A. V. Volyar, E. G. Abramochkin, M. V. Bretsko et al., Journal of Optics 10, 727 (2023). https://doi.org/10.3390/photonics10070727

work page doi:10.3390/photonics10070727 2023

[25] [25]

Ndagano, N

B. Ndagano, N. Mphuthi, G. Milione, et al., Opt. Lett. 42, 4175 (2017). https://doi.org/10.1364/OL.42.004175

work page doi:10.1364/ol.42.004175 2017

[26] [26]

Khorin, Journal of Physics: Conference Series 1096, 012104 (2018)

P. Khorin, Journal of Physics: Conference Series 1096, 012104 (2018). https://doi.org/10.1088/1742-6596/1096/1/012104

work page doi:10.1088/1742-6596/1096/1/012104 2018

[27] [27]

Gibson, M

G. Gibson, M. Graham, J. Courtial, et al., Optics express 12, 5448 (2004). https://doi.org/10.1364/OPEX.12.005448

work page doi:10.1364/opex.12.005448 2004

[28] [28]

J. Wang, J. Yang, I. Fazal, et al., Nat. Photonics 6, 488 (2012). https://doi.org/10.1038/nphoton.2012.138

work page doi:10.1038/nphoton.2012.138 2012

[29] [29]

A. E. Willner, H. Huang, Y. Yan, et al., Adv. Opt. Photonics 7, 66 (2015). https://doi.org/10.1364/AOP.7.000066

work page doi:10.1364/aop.7.000066 2015

[30] [30]

D. D. Reshetnikov, A. L. Sokolov, E. A. Vashukevich, et al., Radiophys Quantum El. 67, 51 (2024). https://doi.org/10.1007/s11141-025-10352-z

work page doi:10.1007/s11141-025-10352-z 2024

[31] [31]

Barros, S

R. Barros, S. Keary, L. Yatcheva, et al., in Remote Sensing of Clouds and the Atmosphere XIX; and Optics in Atmospheric Propagation and Adaptive Systems XVII, A. Comerón, E. I. Kassianov, K. Schäfer, et al., Eds., Proc. SPIE 9242, 92421L (2014). https://doi.org/10.1117/12.2070694

work page doi:10.1117/12.2070694 2014

[32] [32]

V. P. Lukin, Phys. Usp. 10, 280 (2021). https://doi.org/10.3367/UFNr.2020.10.038849

work page doi:10.3367/ufnr.2020.10.038849 2021

[33] [33]

Lukin, P

V. Lukin, P. Konyaev, and V. Sennikov, Applied Optics 51, 176 (2012). https://doi.org/10.1364/AO.51.000C84

work page doi:10.1364/ao.51.000c84 2012

[34] [34]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, Quantum Electron. 46, 729 (2016). https://doi.org/10.1070/QEL16088

work page doi:10.1070/qel16088 2016

[35] [35]

V. P. Aksenov and C. E. Pogutsa, Applied optics 51, 7262 (2012). https://doi.org/10.1364/AO.51.007262

work page doi:10.1364/ao.51.007262 2012

[36] [36]

D. D. Reshetnikov, T. K. Korol, E. V. Malyutina, et al., Optics and Spectroscopy 132, 705 (2024). https://doi.org/10.61011/EOS.2024.07.59649.6774-24

work page doi:10.61011/eos.2024.07.59649.6774-24 2024

[37] [37]

M. S. Kirilenko, P. A. Khorin, and A. P. Porfirev, in CEUR Workshop Proceedings, 1638, 66 (2016). https://doi.org/10.18287/1613-0073-2016-1638-66-75

work page doi:10.18287/1613-0073-2016-1638-66-75 2016

[38] [38]

A. Wada, T. Ohtani, Y. Miyamoto, et al., Journal of the Optical Society of America A. 22, 2746 (2005). https://doi.org/10.1364/JOSAA.22.002746

work page doi:10.1364/josaa.22.002746 2005

[39] [39]

B. C. Platt, R. Shack, Journal of Refractive Surgery 17, S573 (2001). https://doi.org/10.3928/1081-597X-20010901-13

work page doi:10.3928/1081-597x-20010901-13 2001

[40] [40]

Q. Yu, Y. Li, M. Qin, X. Hao, Acta Optica Sinica, 45, 2100001 (2025). https://doi.org/10.3788/AOS251187 ` 10

work page doi:10.3788/aos251187 2025

[41] [41]

Wünsche, Generalized Zernike or disc polynomials 174, 135 (2005)

A. Wünsche, Generalized Zernike or disc polynomials 174, 135 (2005). https://doi.org/10.1016/j.cam.2004.04.004

work page doi:10.1016/j.cam.2004.04.004 2005

[42] [42]

Solimeno, B

S. Solimeno, B. Crosignani, and P. D. Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Elsevier Science, The University of California, 1986)

work page 1986

[43] [43]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Optics Communications 119, 604 (1995). https://doi.org/10.1016/0030-4018(95)00267-C

work page doi:10.1016/0030-4018(95)00267-c 1995

[44] [44]

Ricci, W

F. Ricci, W. Lo ̈ffler, and M. P. van Exter, Optics Express 20, 22961 (2012). https://doi.org/10.1364/OE.20.022961

work page doi:10.1364/oe.20.022961 2012

[45] [45]

S. Zhao, L. Wang, L. Zou, et al., Optics Communications 376, 92 (2016). https://doi.org/10.1016/j.optcom.2016.04.075

work page doi:10.1016/j.optcom.2016.04.075 2016

[46] [46]

L. Li, R. Zhang, Z. Zhao, et al., Sci. Rep. 7, 17427 (2017). https://doi.org/10.1038/s41598- 017-17580-y

work page doi:10.1038/s41598- 2017

[47] [47]

Torner, J

L. Torner, J. P. Torres, and S. Carrasco, Sci. Rep. 13, 873 (2005). https://doi.org/10.1364/OPEX.13.000873

work page doi:10.1364/opex.13.000873 2005

[48] [48]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, et al., Opt. Commun. 103, 422 (1993). https://doi.org/10.1016/0030-4018(93)90168-5

work page doi:10.1016/0030-4018(93)90168-5 1993

[49] [49]

Molina-Terriza, J

G. Molina-Terriza, J. Recolons, J. P. Torres, et al., Phys. Rev. Lett. 87, 023902 (2001). https://doi.org/10.1103/PhysRevLett.87.023902

work page doi:10.1103/physrevlett.87.023902 2001

[50] [50]

Koornwinder, R

T. Koornwinder, R. Wong, R. Koekoek, et al., NIST handbook of mathematical functions (Cambridge University Press, Cambridge, 2010)

work page 2010

[51] [51]

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, Elsevier/Academic Press, Amsterdam, 2007)

work page 2007

[52] [52]

Arfken, The incomplete gamma function and related functions (3rd ed., Academic Press, Orlando, FL, 1985)

G. Arfken, The incomplete gamma function and related functions (3rd ed., Academic Press, Orlando, FL, 1985). ` 11 TABLES Table 1. Zernike expansion coefficients 𝑆𝑝,ℓ (𝑛=|ℓ|) as a function of radial index 𝑝 and azimuthal index ℓ. 𝑆𝑝,ℓ (𝑛=|ℓ|) |ℓ|=0 |ℓ|=1 |ℓ|=2 |ℓ|=3 |ℓ|=4 |ℓ|=5 |ℓ|=6 |ℓ|=7 |ℓ|=8 |ℓ|=9 p = 0 0.447 0.187 0.070 0.023 0.007 0.002 0.001 0.000 0...

work page 1985