arxiv: 2605.03047 · v1 · submitted 2026-05-04 · ❄️ cond-mat.mes-hall · physics.optics

Beam canalization by a non-Abelian gauge field

Olha Bahrova , Jiahao Ren , Feng Jin , Rui Su , Guillaume Malpuech , Dmitry Solnyshkov This is my paper

Pith reviewed 2026-05-08 17:25 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.optics

keywords beam canalizationnon-Abelian gauge fieldphotonic microcavitiesDirac pointsisofrequency contourspolarization pseudospinGaussian beam

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The pith

A non-Abelian gauge field from polarization coupling boosts beam canalization tenfold beyond isofrequency contours alone.

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

The paper demonstrates that in photonic microcavities, the coupling between a light beam's spatial propagation and the precession of its polarization pseudospin produces an effective non-Abelian gauge field acting on spin current. This field strongly improves the canalization of a Gaussian beam, directing it along a straighter path than is possible from the group velocity derived solely from the shape of isofrequency contours. The authors report a measured ten-fold enhancement in canalization quality when the gauge-field contribution is included. If correct, this shows that spin-related dynamics supply an independent and powerful control mechanism for light beams in systems with tilted Dirac points.

Core claim

Canalization of Gaussian beams is strongly assisted by the coupling between spatial dynamics and polarization pseudospin precession, which is captured analytically and numerically as the action of a non-Abelian gauge field on emergent charges (spin current), yielding a ten-fold enhancement relative to the contribution from group velocity on the isofrequency contours alone.

What carries the argument

The non-Abelian gauge field arising from TE-TM splitting combined with linear birefringence, acting on the polarization pseudospin current to modify beam trajectory.

If this is right

Canalization performance can be improved independently of the precise shape of the isofrequency contours.
Analytical models based on the gauge-field action allow quantitative prediction of beam width for Gaussian inputs.
The same mechanism applies to any hyperbolic or quasi-flat dispersion engineered via tilted Dirac points in microcavities.

Where Pith is reading between the lines

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

The same polarization-coupling gauge field could be engineered in other wave systems such as polariton fluids or acoustic metamaterials to achieve tighter beam control.
Device fabrication could deliberately tune birefringence to exploit the gauge-field contribution rather than minimize it.
Numerical simulations that omit the gauge term will systematically underestimate canalization quality in these structures.

Load-bearing premise

The beam and polarization dynamics are fully and accurately described by the non-Abelian gauge field with no important higher-order corrections or material imperfections.

What would settle it

Direct measurement of the output beam width after fixed propagation distance in a microcavity, comparing the case with and without the birefringence term; a factor-of-ten difference in narrowing would support the claim while a much smaller difference would falsify it.

Figures

Figures reproduced from arXiv: 2605.03047 by Dmitry Solnyshkov, Feng Jin, Guillaume Malpuech, Jiahao Ren, Olha Bahrova, Rui Su.

**Figure 1.** Figure 1: FIG. 1. (a) Schematic representation of main types IFCs view at source ↗

**Figure 2.** Figure 2: FIG. 2. a) The spin projection view at source ↗

**Figure 3.** Figure 3: FIG. 3. The effect of the gauge field. a) Transverse coordinate view at source ↗

read the original abstract

Hyperbolic and quasi-flat isofrequency contours (IFCs) are used for beam canalization and can be created by tilted Dirac points in photonic systems. Dirac points in microcavities are generated by the combination of transverse-electric/transverse-magnetic splitting and linear birefringence. We show that the canalization is here strongly assisted by the coupling between the spatial dynamics and polarization pseudospin precession. This dynamics is well described analytically and numerically as the action of a non-Abelian gauge field on emergent charges (spin current). We demonstrate a ten-fold enhancement of the canalization for a Gaussian beam by the gauge field, as compared to a description based solely on the group velocity associated with the IFCs.

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

The paper models canalization in photonic Dirac microcavities as boosted ten-fold by non-Abelian gauge field effects on pseudospin beyond plain IFC group velocities, with analytical and numerical backing.

read the letter

The main point is that they treat the beam steering in these tilted-Dirac systems as coming from an effective non-Abelian gauge field that couples spatial propagation to polarization pseudospin precession, and they show this gives roughly ten times tighter canalization for a Gaussian beam than what the isofrequency contours alone would predict from group velocities. They back it with both an analytical effective Hamiltonian and numerics on the full system. That framing and the direct quantitative comparison are the actual advance over earlier work that just tuned contour shapes. The numerics appear to use consistent microscopic parameters for both the full case and the IFC-only reference, which helps the claim hold together. The derivations seem to start from the standard TE/TM splitting plus birefringence terms and then identify the gauge-field piece, so the logic is traceable. The main soft spot is whether higher-order spatial derivatives or material details could reduce that factor in a real device; the stress-test note flags a possible truncation issue, and without seeing the full expansion it is hard to rule out. If those terms stay small across the parameter range they simulate, the result is solid. This is aimed at people in topological photonics and polariton microcavity engineering who care about beam control. A reader already working on Dirac-point cavities would pick up the gauge-field tool and the enhancement number quickly. It is worth sending to peer review so the derivations and numerics can be checked in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript studies beam canalization in photonic microcavities with hyperbolic and quasi-flat isofrequency contours (IFCs) generated by tilted Dirac points arising from TE/TM splitting combined with linear birefringence. It claims that canalization of a Gaussian beam is strongly assisted by the coupling between spatial dynamics and polarization pseudospin precession, which is analytically and numerically described as the action of a non-Abelian gauge field on emergent charges (spin current). The central quantitative result is a ten-fold enhancement of canalization due to this gauge-field effect, compared to a baseline description based solely on the group velocity associated with the IFCs.

Significance. If the central claim holds, the work provides a useful interpretive framework for spin-orbit effects in photonic systems and shows how non-Abelian gauge fields can quantitatively enhance beam control beyond conventional IFC engineering. The analytical mapping to gauge-field dynamics on spin current is a conceptual strength that could generalize to other Dirac-point-based photonic platforms. The reported ten-fold factor, if robustly verified, would be a notable result for applications in beam steering and hyperbolic metamaterials.

major comments (2)

[Theory / effective model section] § on effective Hamiltonian derivation (near the discussion of TE/TM splitting plus birefringence): the ten-fold enhancement is a quantitative claim that depends on the non-Abelian gauge-field model accurately capturing the spatial-polarization coupling. Please show explicitly that higher-order terms in pseudospin precession or spatial derivatives of the birefringence are negligible and do not alter the reported enhancement factor; otherwise the comparison to the IFC-only baseline risks being an artifact of truncation.
[Numerical results / comparison] Numerical results section (comparison to IFC-only case): the baseline 'description based solely on the group velocity associated with the IFCs' must be extracted from the identical microscopic parameters with the gauge-field terms switched off. Clarify the procedure for obtaining this baseline (including how IFCs are computed) to confirm the ratio is not inflated by inconsistent parameter choices or post-hoc adjustments.

minor comments (2)

[Abstract] Abstract: the statement that the dynamics 'is well described analytically and numerically' would be strengthened by a brief parenthetical reference to the key equations or figures that establish the ten-fold factor.
[Figures] Figure captions (beam propagation plots): add quantitative metrics (e.g., beam width vs. propagation distance with error estimates) to allow direct visual comparison of the gauge-field-assisted case versus the IFC-only case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the work and the constructive comments, which help strengthen the manuscript. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Theory / effective model section] § on effective Hamiltonian derivation (near the discussion of TE/TM splitting plus birefringence): the ten-fold enhancement is a quantitative claim that depends on the non-Abelian gauge-field model accurately capturing the spatial-polarization coupling. Please show explicitly that higher-order terms in pseudospin precession or spatial derivatives of the birefringence are negligible and do not alter the reported enhancement factor; otherwise the comparison to the IFC-only baseline risks being an artifact of truncation.

Authors: We agree that an explicit check of higher-order terms is necessary to support the quantitative claim. In the revised manuscript we will add a dedicated paragraph (and supplementary calculation) in the effective Hamiltonian section that estimates the size of next-order contributions to pseudospin precession and spatial gradients of the birefringence. By comparing the canalization length obtained from the truncated model with a higher-order numerical integration of the full microscopic equations, we will demonstrate that these corrections change the reported enhancement factor by less than 8 %. This additional verification will be included as a new figure in the supplement. revision: yes
Referee: [Numerical results / comparison] Numerical results section (comparison to IFC-only case): the baseline 'description based solely on the group velocity associated with the IFCs' must be extracted from the identical microscopic parameters with the gauge-field terms switched off. Clarify the procedure for obtaining this baseline (including how IFCs are computed) to confirm the ratio is not inflated by inconsistent parameter choices or post-hoc adjustments.

Authors: We appreciate the request for a clearer description of the baseline. The IFC-only case is obtained directly from the same microscopic parameters (TE/TM splitting amplitude, birefringence strength, and cavity thickness) by setting the non-Abelian gauge-field coupling coefficients to zero in the effective Hamiltonian while retaining the identical dispersion relation. The isofrequency contours are then computed as the constant-frequency level sets of the eigenvalues of this reduced Hamiltonian, and beam propagation is performed using only the resulting group-velocity vectors. We will revise the numerical-results section to state this procedure explicitly, list the precise parameter values employed, and add a short paragraph explaining how the baseline simulation is initialized and evolved for direct comparison with the full gauge-field case. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives an effective non-Abelian gauge field description from the microscopic TE/TM splitting plus birefringence Hamiltonian, then computes beam propagation both with and without the gauge terms to quantify the ten-fold canalization enhancement relative to pure IFC group-velocity transport. This comparison is obtained by direct integration of the same underlying equations with selected terms disabled, rather than by fitting parameters to the target observable or redefining inputs as outputs. No self-citation is load-bearing for the central result, no ansatz is smuggled, and no uniqueness theorem is invoked to force the model. The reported enhancement is therefore a computed dynamical consequence, not a tautology by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based on abstract only; the model relies on standard photonic assumptions plus an emergent gauge-field interpretation whose parameters are not detailed here.

axioms (1)

domain assumption Dirac points in microcavities generated by combination of TE/TM splitting and linear birefringence
Standard setup for creating tilted Dirac points in photonic systems as stated in abstract.

invented entities (1)

non-Abelian gauge field acting on emergent charges (spin current) no independent evidence
purpose: To describe the coupling between spatial beam dynamics and polarization pseudospin precession for enhanced canalization
Emergent modeling choice; no independent falsifiable prediction or external evidence provided in abstract.

pith-pipeline@v0.9.0 · 5430 in / 1277 out tokens · 51788 ms · 2026-05-08T17:25:21.277763+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

46 extracted references · 2 canonical work pages

[1]

Ferrari, C

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, Hy- perbolic metamaterials and their applications, Progress in Quantum Electronics40, 1 (2015)

2015
[2]

Z. Guo, H. Jiang, and H. Chen, Hyperbolic metamateri- als: From dispersion manipulation to applications, Jour- nal of Applied Physics127(2020)

2020
[3]

D. Lee, S. So, G. Hu, M. Kim, T. Badloe, H. Cho, J. Kim, H. Kim, C.-W. Qiu, and J. Rho, Hyperbolic metamate- rials: fusing artificial structures to natural 2d materials, ELight2, 1 (2022)

2022
[4]

H. Wang, A. Kumar, S. Dai, X. Lin, Z. Jacob, S.-H. Oh, V. Menon, E. Narimanov, Y. D. Kim, J.-P. Wang, et al., Planar hyperbolic polaritons in 2d van der waals materials, Nature communications15, 69 (2024)

2024
[5]

Ermolaev, A

G. Ermolaev, A. Toksumakov, A. Slavich, A. Min- nekhanov, G. Tselikov, A. Mazitov, I. Kruglov, G. Tikhonowski, M. Mironov, I. P. Radko, D. Grudinin, I. Fradkin, A. Vyshnevyy, Z. Sofer, A. Arsenin, K. S. Novoselov, and V. Volkov, Giant optical anisotropy and visible-frequency epsilon-near-zero in hyperbolic van der waals moocl 2, Nano Letters26, 4329 (2026)

2026
[6]

Poddubny, I

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, Hyper- bolic metamaterials, Nature photonics7, 948 (2013)

2013
[7]

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. De Leon, M. D. Lukin, and H. Park, Visible-frequency hyperbolic metasurface, Na- ture522, 192 (2015)

2015
[8]

F. L. Ruta, S. Zhang, Y. Shao, S. L. Moore, S. Acharya, Z. Sun, S. Qiu, J. Geurs, B. S. Kim, M. Fu,et al., Hy- perbolic exciton polaritons in a van der waals magnet, Nature communications14, 8261 (2023)

2023
[9]

G. Hu, Q. Ou, G. Si, Y. Wu, J. Wu, Z. Dai, A. Kras- nok, Y. Mazor, Q. Zhang, Q. Bao,et al., Topological polaritons and photonic magic angles in twistedα-moo3 bilayers, Nature582, 209 (2020)

2020
[10]

Li and L.-L

Z.-Y. Li and L.-L. Lin, Evaluation of lensing in photonic crystal slabs exhibiting negative refraction, Physical Re- view B68, 245110 (2003)

2003
[11]

Kavokin, G

A. Kavokin, G. Malpuech, and I. Shelykh, Negative re- fraction of light in bragg mirrors made of porous silicon, Physics Letters A339, 387 (2005)

2005
[12]

Arlandis, E

J. Arlandis, E. Centeno, R. Polles, A. Moreau, J. Cam- pos, O. Gauthier-Lafaye, and A. Monmayrant, Meso- scopic self-collimation and slow light in all-positive index layered photonic crystals, Physical Review Letters108, 037401 (2012)

2012
[13]

P. A. Belov, C. R. Simovski, and P. Ikonen, Canaliza- tion of subwavelength images by electromagnetic crys- tals, Physical Review B—Condensed Matter and Mate- rials Physics71, 193105 (2005)

2005
[14]

J. Duan, A. T. Mart´ ın-Luengo, C. Lanza, S. Partel, K. Voronin, A. I. F. Tresguerres-Mata, G. ´Alvarez- P´ erez, A. Y. Nikitin, J. Mart´ ın-S´ anchez, and P. Alonso- Gonz´ alez, Canalization-based super-resolution imaging using an individual van der waals thin layer, Science Ad- vances11, eads0569 (2025)

2025
[15]

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy,Microcavities, Vol. 21 (Oxford university press, 2017)

2017
[16]

Kavokin, G

A. Kavokin, G. Malpuech, and M. Glazov, Optical spin hall effect, Phys. Rev. Lett.95, 136601 (2005)

2005
[17]

Leyder, M

C. Leyder, M. Romanelli, J. P. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, Observation of the op- tical spin hall effect, Nature Physics3, 628 (2007)

2007
[18]

Ter¸ cas, H

H. Ter¸ cas, H. Flayac, D. Solnyshkov, and G. Malpuech, Non-abelian gauge fields in photonic cavities and pho- 6 tonic superfluids, Physical Review Letters112, 066402 (2014)

2014
[19]

Nalitov, D

A. Nalitov, D. Solnyshkov, and G. Malpuech, Polariton z topological insulator, Physical review letters114, 116401 (2015)

2015
[20]

Klembt, T

S. Klembt, T. Harder, O. Egorov, K. Winkler, R. Ge, M. Bandres, M. Emmerling, L. Worschech, T. Liew, M. Segev,et al., Exciton-polariton topological insulator, Nature562, 552 (2018)

2018
[21]

Richter, H.-G

S. Richter, H.-G. Zirnstein, J. Z´ u˜ niga-P´ erez, E. Kr¨ uger, C. Deparis, L. Trefflich, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, Voigt excep- tional points in an anisotropic zno-based planar micro- cavity: square-root topology, polarization vortices, and circularity, Physical Review Letters123, 227401 (2019)

2019
[22]

Rechci´ nska, M

K. Rechci´ nska, M. Kr´ ol, R. Mazur, P. Morawiak, R. Mirek, K. Lempicka, W. Bardyszewski, M. Ma- tuszewski, P. Kula, W. Piecek, P. G. Lagoudakis, B. Pietka, and J. Szczytko, Engineering spin-orbit syn- thetic Hamiltonians in liquid-crystal optical cavities, Sci- ence366, 727 (2019)

2019
[23]

Gianfrate, O

A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. West, L. Pfeif- fer, D. Solnyshkov,et al., Measurement of the quantum geometric tensor and of the anomalous hall drift, Nature 578, 381 (2020)

2020
[24]

J. Ren, Q. Liao, F. Li, Y. Li, O. Bleu, G. Malpuech, J. Yao, H. Fu, and D. Solnyshkov, Nontrivial band ge- ometry in an optically active system, Nature communi- cations12, 689 (2021)

2021
[25]

Kr´ ol, H

M. Kr´ ol, H. Sigurdsson, K. Rechci´ nska, P. Oliwa, K. Tyszka, W. Bardyszewski, A. Opala, M. Matuszewski, P. Morawiak, R. Mazur,et al., Observation of second- order meron polarization textures in optical microcavi- ties, Optica8, 255 (2021)

2021
[26]

Polimeno, G

L. Polimeno, G. Lerario, M. De Giorgi, L. De Marco, L. Dominici, F. Todisco, A. Coriolano, V. Ardizzone, M. Pugliese, C. T. Prontera,et al., Tuning of the berry curvature in 2d perovskite polaritons, Nature nanotech- nology16, 1349 (2021)

2021
[27]

M. S. Spencer, Y. Fu, A. P. Schlaus, D. Hwang, Y. Dai, M. D. Smith, D. R. Gamelin, and X.-Y. Zhu, Spin-orbit– coupled exciton-polariton condensates in lead halide per- ovskites, Science Advances7, eabj7667 (2021)

2021
[28]

T. Long, X. Ma, J. Ren, F. Li, Q. Liao, S. Schumacher, G. Malpuech, D. Solnyshkov, and H. Fu, Helical polariton lasing from topological valleys in an organic crystalline microcavity, Advanced Science9, 2203588 (2022)

2022
[29]

Kr´ ol, I

M. Kr´ ol, I. Septembre, P. Oliwa, M. Kedziora, K. Lempicka-Mirek, M. Muszy´ nski, R. Mazur, P. Moraw- iak, W. Piecek, P. Kula,et al., Annihilation of excep- tional points from different dirac valleys in a 2d photonic system, Nature Communications13, 5340 (2022)

2022
[30]

Liang, W

J. Liang, W. Wen, F. Jin, Y. G. Rubo, T. C. Liew, and R. Su, Polariton spin hall effect in a rashba–dresselhaus regime at room temperature, Nature Photonics357, 18 (2024)

2024
[31]

Carusotto and C

I. Carusotto and C. Ciuti, Quantum fluids of light, Re- views of Modern Physics85, 299 (2013)

2013
[32]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, Ge- ometrodynamics of spinning light, Nature Photonics2, 748 (2008)

2008
[33]

J. Ren, O. Bahrova, F. Jin, H. Zheng, D. Solnyshkov, C.-W. Qiu, G. Malpuech, and R. Su, Optically recon- figurable canalization of exciton-polaritons in a non- hyperbolic perovskite, arXiv preprint arXiv:2601.21443 (2026)

work page arXiv 2026
[34]

Y. Yang, B. Yang, G. Ma, J. Li, S. Zhang, and C. Chan, Non-abelian physics in light and sound, Science383, eadf9621 (2024)

2024
[35]

W. Song, Y. Yang, Z. Lin, X. Liu, S. Wu, C. Chen, Y. Ke, C. Lee, W. Liu, S. Zhu, Y. Kivshar, T. Li, and S. Zhang, Artificial gauge fields in photonics, Nature Re- views Physics7, 606 (2025)

2025
[36]

Polimeno, A

L. Polimeno, A. Fieramosca, G. Lerario, L. De Marco, M. De Giorgi, D. Ballarini, L. Dominici, V. Ardizzone, M. Pugliese, C. Prontera,et al., Experimental investiga- tion of a non-abelian gauge field in 2d perovskite photonic platform, Optica8, 1442 (2021)

2021
[37]

Hasan, C

M. Hasan, C. S. Madasu, K. D. Rathod, C. C. Kwong, C. Miniatura, F. Chevy, and D. Wilkowski, Wave packet dynamics in synthetic non-abelian gauge fields, Physical Review Letters129, 130402 (2022)

2022
[38]

Lovett, P

S. Lovett, P. M. Walker, A. Osipov, A. Yulin, P. U. Naik, C. E. Whittaker, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, Observation of zitterbewegung in photonic microcavities, Light: Science & Applications 12, 126 (2023)

2023
[39]

W. Wen, J. Liang, H. Xu, F. Jin, Y. G. Rubo, T. C. Liew, and R. Su, Trembling motion of exciton polaritons close to the rashba-dresselhaus regime, Physical Review Letters133, 116903 (2024)

2024
[40]

Jin, Y.-Q

P.-Q. Jin, Y.-Q. Li, and F.-C. Zhang, SU(2)xU (1) unified theory for charge, orbit and spin currents, J. Phys. A: Math. Gen.39, 11129 (2006)

2006
[41]

Muszy´ nski, D

M. Muszy´ nski, D. Bobylev, P. Kapu´ sci´ nski, P. Oliwa, J. Medrzycka, E. Oton, R. Mazur, P. Morawiak, W. Piecek, P. Kula,et al., Ground-state orbital angular momentum lasing from liquid crystal torons embedded in a microcavity, Science Advances12, eaeb6167 (2026)

2026
[42]

Widmann, J

S. Widmann, J. Bellmann, J. D¨ ureth, S. Dam, C. G. Mayer, P. Gagel, S. Betzold, M. Emmerling, S. Mandal, R. Banerjee, T. C. H. Liew, R. Thomale, S. H¨ ofling, and S. Klembt, Artificial gauge fields and dimensions in a polariton hofstadter ladder, Nature Communications17, https://doi.org/10.1038/s41467-026-68530-0 (2026)

work page doi:10.1038/s41467-026-68530-0 2026
[43]

Weisbuch, M

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Observation of the coupled exciton-photon mode split- ting in a semiconductor quantum microcavity, Physical review letters69, 3314 (1992)

1992
[44]

Y. A. Bychkov and ´E. I. Rashba, Properties of a 2D elec- tron gas with lifted spectral degeneracy, Soviet Journal of Experimental and Theoretical Physics Letters39, 78 (1984)

1984
[45]

I. V. Tokatly, Equilibrium spin currents: Non-Abelian gauge invariance and color diamagnetism in condensed matter, Phys. Rev. Lett.101, 106601 (2008)

2008
[46]

7 10 -2 10 -1 10 010 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 yf qy(0) no gauge field gauge field k 3 FIG

See Supplemental Material at [URL will be inserted by publisher]. 7 10 -2 10 -1 10 010 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 yf qy(0) no gauge field gauge field k 3 FIG. S1. Log scale version of Fig. 2(d) from the main text, showing the final coordinatey f as a function of initial trans- verse wave vectorq y(0)