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arxiv: 2211.09884 · v2 · submitted 2022-11-17 · ⚛️ physics.optics

Band Inversion Flips the Winding of Bound States in the Continuum

Paul Bouteyre , Dung Xuan Nguyen , Lo\"ic Malgrey , Zhiyi Yuan , Guillaume Gachon , Taha Benyattou , Xavier Letartre , Pierre Viktorovitch
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S\'egol\`ene Callard Guangwei Hu Shanhui Fan Lydie Ferrier Hai Son Nguyen
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Pith reviewed 2026-05-24 11:12 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords bound states in the continuumband inversionwinding numberpolarization singularitiesphotonic slabsmetasurfacesnon-Hermitian photonics
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The pith

Band inversion at a band-edge bound state in the continuum reverses its far-field polarization map and flips the winding number.

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

The paper establishes that band inversion flips the winding of bound states in the continuum. Using a two-band theory for open periodic photonic structures, it proves that inversion at the band edge reverses the local polarization pattern, changing w_BIC to -w_BIC without defects moving in momentum space. This challenges the view of winding as a robust topological label under smooth deformations. Verification comes from a tunable subwavelength grating with polarization tomography and numerical checks in rectangular and triangular lattices. The result highlights band inversion as a mechanism for controlling polarization-singularity topology in non-Hermitian photonic bands.

Core claim

Bound states in the continuum appear as polarization singularities in momentum space with an integer winding number that is treated as robust under smooth deformations. A band inversion at a band-edge BIC reverses the local far-field polarization map and flips the BIC winding w_BIC to -w_BIC without any defect dynamics in momentum space. This is proven using a general two-band theory and verified experimentally in a tunable grating and numerically in multiband lattices.

What carries the argument

Two-band theory of open periodic photonic structures that connects band inversion to reversal of the far-field polarization map at BICs.

If this is right

  • The winding number of a BIC is not invariant under band inversion.
  • Polarization tomography in a tunable subwavelength grating directly images the winding flip.
  • The effect appears in multiband rectangular and triangular photonic lattices.
  • Band inversion governs polarization-singularity topology in non-Hermitian photonic band structures.

Where Pith is reading between the lines

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

  • Designers of photonic devices could use band inversion to switch the topological character of BICs without altering the lattice geometry.
  • This mechanism might extend to other wave systems where band inversions occur, such as acoustics or quantum mechanics.
  • Further study could explore whether similar flips happen in higher-dimensional or non-periodic structures.

Load-bearing premise

The two-band theory of open periodic photonic structures is sufficiently general to capture the polarization reversal under band inversion.

What would settle it

Direct measurement in the tunable grating showing no polarization map reversal upon band inversion would disprove the winding flip.

Figures

Figures reproduced from arXiv: 2211.09884 by Dung Xuan Nguyen, Guangwei Hu, Guillaume Gachon, Hai Son Nguyen, Lo\"ic Malgrey, Lydie Ferrier, Paul Bouteyre, Pierre Viktorovitch, S\'egol\`ene Callard, Shanhui Fan, Taha Benyattou, Xavier Letartre, Zhiyi Yuan.

Figure 1
Figure 1. Figure 1: (a) Sketch of the sample: photonic lattices consisting of gold gratings of period [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a to d) Mapping of the farfield polarization orientation [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Three-dimensional vector M which is composed of the polarization vector  = (Ex, Ey)/E0, and the band index U as the vertical coordinate. The pseudo-spin m is then defined by m = M/||M||. (b,c) The topological transition from antimeron texture (b) to (c) meron texture of the pseudo-spin m through the band inversion. The upper panels of (b,c) show the texture of m in the momentum space, while the lower … view at source ↗
read the original abstract

Bound states in the continuum (BICs) in photonic slabs and metasurfaces appear as polarization singularities in momentum space, characterized by an integer winding number. This winding is widely treated as a robust topological label, preserved under smooth deformations of the structure. Here we show that this robustness fails under band inversion. Using a general two-band theory of open periodic photonic structures, we prove that a band inversion at a band-edge BIC reverses the local far-field polarization map and flips the BIC winding, $w_{\rm BIC}\to -w_{\rm BIC}$, \emph{without} any defect dynamics in momentum space. We verify the prediction in a tunable subwavelength grating, where polarization tomography directly images the reversal, and confirm it numerically in multiband rectangular and triangular photonic lattices. Band inversion thus emerges as a key mechanism governing polarization-singularity topology in non-Hermitian photonic band structures.

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.

Referee Report

1 major / 1 minor

Summary. The paper claims that a band inversion at a band-edge bound state in the continuum (BIC) in photonic slabs and metasurfaces reverses the local far-field polarization map around the BIC, thereby flipping its integer winding number w_BIC to -w_BIC, without any defect motion in momentum space. This is derived from a general two-band theory of open periodic photonic structures that maps the inversion directly to polarization reversal; the prediction is verified by polarization tomography in a tunable subwavelength grating and by numerical simulations in rectangular and triangular multiband photonic lattices.

Significance. If the central claim holds, the result identifies band inversion as a mechanism that can alter the topological winding of polarization singularities in non-Hermitian photonic band structures, contrary to the usual assumption that winding is preserved under smooth deformations. The combination of an analytic two-band derivation, direct experimental imaging of the polarization texture, and numerical confirmation across lattice types provides a concrete, falsifiable route to controlling BIC topology. Credit is due for the explicit experimental tomography and the multiband numerical checks.

major comments (1)
  1. [two-band theory section / abstract] Abstract and the section presenting the general two-band theory: the proof that far-field Stokes parameters (and thus the winding) are determined solely by the 2x2 non-Hermitian eigenvectors assumes negligible leakage from higher bands. No explicit bound or detuning criterion is supplied showing that other bands remain sufficiently detuned throughout the inversion sweep; the numerical checks in rectangular/triangular lattices are cited but do not report the minimum detuning or the residual coupling strength to higher bands during the parameter sweep.
minor comments (1)
  1. [abstract] The abstract mentions 'post-hoc tuning of the grating' but does not quantify how this tuning affects the extracted winding or the error bars on the polarization map.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [two-band theory section / abstract] Abstract and the section presenting the general two-band theory: the proof that far-field Stokes parameters (and thus the winding) are determined solely by the 2x2 non-Hermitian eigenvectors assumes negligible leakage from higher bands. No explicit bound or detuning criterion is supplied showing that other bands remain sufficiently detuned throughout the inversion sweep; the numerical checks in rectangular/triangular lattices are cited but do not report the minimum detuning or the residual coupling strength to higher bands during the parameter sweep.

    Authors: We agree that the two-band derivation implicitly assumes higher bands remain sufficiently detuned. In the revised manuscript we will add an explicit perturbative criterion: the interband detuning must exceed several times the radiative leakage rate of the BIC mode (derived from the non-Hermitian two-band Hamiltonian). We will also report the minimum detuning and an upper bound on residual higher-band coupling strength extracted from the rectangular and triangular lattice simulations across the inversion sweep. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from independent two-band model

full rationale

The paper's central claim—that band inversion at a band-edge BIC reverses the far-field polarization map and thus flips w_BIC—is derived from an explicit two-band non-Hermitian Hamiltonian whose eigenvectors determine the local Stokes parameters. This mapping is not self-definitional: the Hamiltonian is introduced as a general effective model for open periodic structures, and its predictions are then verified both experimentally in a tunable grating and numerically in multiband rectangular/triangular lattices where higher bands remain detuned. No parameters are fitted to the target winding and then relabeled as a prediction, no uniqueness theorem is imported via self-citation, and no ansatz is smuggled in. The derivation therefore remains self-contained against external benchmarks rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the generality of the two-band model for open periodic structures and on the experimental ability to induce band inversion while keeping the BIC at the band edge.

axioms (1)
  • domain assumption The two-band theory of open periodic photonic structures is general and captures the essential physics of band inversion at BICs.
    Invoked explicitly as the basis for proving the winding reversal.

pith-pipeline@v0.9.0 · 5736 in / 1223 out tokens · 23530 ms · 2026-05-24T11:12:47.720939+00:00 · methodology

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    V on Neumann and E

    J. V on Neumann and E. Wigner, Über merkwürdige diskrete Eigenwerte, Z. Phys 30, 465 (1929)

    work page 1929

  2. [2]

    Friedrich and D

    H. Friedrich and D. Wintgen, Interfering resonances and bound states in the continuum, Physical Review A 32, 3231 (1985)

    work page 1985

  3. [3]

    Friedrich and D

    H. Friedrich and D. Wintgen, Physical realization of bound states in the continuum, Physical Review A 31, 3964 (1985)

    work page 1985

  4. [4]

    L. S. Cederbaum, R. S. Friedman, V . M. Ryaboy, and N. Moi- seyev, Conical Intersections and Bound Molecular States Em- bedded in the Continuum, Physical Review Letters 90, 013001 (2003)

    work page 2003

  5. [5]

    Thomas, M

    R. Thomas, M. Chilcott, E. Tiesinga, A. B. Deb, and N. Kjær- gaard, Observation of bound state self-interaction in a nano-eV atom collider, Nature Communications 9, 4895 (2018)

    work page 2018

  6. [6]

    P. J. Cobelli, V . Pagneux, A. Maurel, and P. Petitjeans, Exper- imental study on water-wave trapped modes, Journal of Fluid Mechanics 666, 445 (2011)

    work page 2011

  7. [7]

    Capasso, C

    F. Capasso, C. Sirtori, J. Faist, D. L. Sivco, S.-N. G. Chu, and A. Y . Cho, Observation of an electronic bound state above a potential well, Nature 358, 565 (1992)

    work page 1992

  8. [8]

    J. U. Nöckel, Resonances in quantum-dot transport, Physical Review B 46, 15348 (1992)

    work page 1992

  9. [9]

    M. L. Ladrón de Guevara and P. A. Orellana, Electronic trans- port through a parallel-coupled triple quantum dot molecule: Fano resonances and bound states in the continuum, Physical Review B 73, 205303 (2006)

    work page 2006

  10. [10]

    C. W. Hsu, B. Zhen, J. Lee, S.-l. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Solja ˇci´c, Observation of trapped light within the radiation continuum, Nature 499, 188 (2013)

    work page 2013

  11. [11]

    Kodigala, T

    A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y . Fainman, and B. Kanté, Lasing action from photonic bound states in contin- uum, Nature 541, 196 (2017)

    work page 2017

  12. [12]

    S. T. Ha, Y . H. Fu, N. K. Emani, Z. Pan, R. M. Bakker, R. Paniagua-Domínguez, and A. I. Kuznetsov, Directional las- ing in resonant semiconductor nanoantenna arrays, Nature Nan- otechnology 13, 1042 (2018)

    work page 2018

  13. [13]

    Huang, C

    C. Huang, C. Zhang, S. Xiao, Y . Wang, Y . Fan, Y . Liu, N. Zhang, G. Qu, H. Ji, J. Han, L. Ge, Y . Kivshar, and Q. Song, Ultrafast control of vortex microlasers, Science 367, 1018 (2020)

    work page 2020

  14. [14]

    Romano, M

    S. Romano, M. Mangini, E. Penzo, S. Cabrini, A. C. De Luca, I. Rendina, V . Mocella, and G. Zito, Ultrasensitive Surface Re- fractive Index Imaging Based on Quasi-Bound States in the Continuum, ACS Nano 14, 15417 (2020)

    work page 2020

  15. [15]

    Tittl, A

    A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.-Y . Choi, D. N. Ne- shev, Y . S. Kivshar, and H. Altug, Imaging-based molecular barcoding with pixelated dielectric metasurfaces, Science 360, 1105 (2018)

    work page 2018

  16. [16]

    B. Zhen, C. W. Hsu, L. Lu, a. D. Stone, and M. Solja ˇci´c, Topo- logical Nature of Optical Bound States in the Continuum, Phys- ical Review Letters 113, 1 (2014)

    work page 2014

  17. [17]

    Zhang, A

    Y . Zhang, A. Chen, W. Liu, C. W. Hsu, B. Wang, F. Guan, X. Liu, L. Shi, L. Lu, and J. Zi, Observation of polarization vor- tices in momentum space, Phys. Rev. Lett.120, 186103 (2018)

    work page 2018

  18. [18]

    H. M. Doeleman, F. Monticone, W. den Hollander, A. Alù, and A. F. Koenderink, Experimental observation of a polarization vortex at an optical bound state in the continuum, Nature Pho- tonics 12, 397 (2018)

    work page 2018

  19. [19]

    Yin and C

    X. Yin and C. Peng, Manipulating light radiation from a topo- logical perspective, Photon. Res. 8, B25 (2020)

    work page 2020

  20. [20]

    J. Jin, X. Yin, L. Ni, M. Solja ˇci´c, B. Zhen, and C. Peng, Topo- logically enabled ultrahigh-Q guided resonances robust to out- of-plane scattering, Nature 574, 501 (2019)

    work page 2019

  21. [21]

    W. Liu, B. Wang, Y . Zhang, J. Wang, M. Zhao, F. Guan, X. Liu, L. Shi, and J. Zi, Circularly polarized states spawning from bound states in the continuum, Phys. Rev. Lett. 123, 116104 (2019)

    work page 2019

  22. [22]

    Yoda and M

    T. Yoda and M. Notomi, Generation and annihilation of topo- logically protected bound states in the continuum and circularly polarized states by symmetry breaking, Phys. Rev. Lett. 125, 053902 (2020)

    work page 2020

  23. [23]

    M. Kang, S. Zhang, M. Xiao, and H. Xu, Merging bound states in the continuum at off-high symmetry points, Phys. Rev. Lett. 126, 117402 (2021)

    work page 2021

  24. [24]

    M. Kang, L. Mao, S. Zhang, M. Xiao, H. Xu, and C. T. Chan, Merging bound states in the continuum by harnessing higher- order topological charges, Light: Science & Applications 11, 228 (2022)

    work page 2022

  25. [25]

    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008)

    work page 2008

  26. [26]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insula- tors, Rev. Mod. Phys. 82, 3045 (2010)

    work page 2010

  27. [27]

    Ozawa, H

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019)

    work page 2019

  28. [28]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin hall effect and topological phase transition in hgte quantum wells, Science 314, 1757 (2006)

    work page 2006

  29. [29]

    Z. Zhu, Y . Cheng, and U. Schwingenschlögl, Band inversion mechanism in topological insulators: A guideline for materials design, Phys. Rev. B 85, 235401 (2012)

    work page 2012

  30. [30]

    Skyrme, A unified field theory of mesons and baryons, Nu- clear Physics 31, 556 (1962)

    T. Skyrme, A unified field theory of mesons and baryons, Nu- clear Physics 31, 556 (1962)

    work page 1962

  31. [31]

    Heinze, K

    S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubet- zka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Sponta- neous atomic-scale magnetic skyrmion lattice in two dimen- sions, Nature Physics 7, 713 (2011)

    work page 2011

  32. [32]

    Ferrier, H

    L. Ferrier, H. S. Nguyen, C. Jamois, L. Berguiga, C. Symonds, J. Bellessa, and T. Benyattou, Tamm plasmon photonic crystals: From bandgap engineering to defect cavity, APL Photonics 4, 106101 (2019)

    work page 2019

  33. [33]

    Lee and R

    S.-G. Lee and R. Magnusson, Band flips and bound-state transi- tions in leaky-mode photonic lattices, Phys. Rev. B 99, 045304 (2019)

    work page 2019

  34. [34]

    Liu and S

    V . Liu and S. Fan, S4 : A free electromagnetic solver for layered periodic structures, Computer Physics Communications 183, 2233 (2012)

    work page 2012

  35. [35]

    C. Guo, M. Xiao, Y . Guo, L. Yuan, and S. Fan, Meron spin tex- tures in momentum space, Phys. Rev. Lett.124, 106103 (2020)

    work page 2020

  36. [36]

    M. Król, H. Sigurdsson, K. Rechci ´nska, P. Oliwa, K. Tyszka, W. Bardyszewski, A. Opala, M. Matuszewski, P. Morawiak, R. Mazur, W. Piecek, P. Kula, P. G. Lagoudakis, B. Pi˛ etka, and J. Szczytko, Observation of second-order meron polariza- tion textures in optical microcavities, Optica 8, 255 (2021)

    work page 2021

  37. [37]

    X. Z. Yu, W. Koshibae, Y . Tokunaga, K. Shibata, Y . Taguchi, N. Nagaosa, and Y . Tokura, Transformation between meron and skyrmion topological spin textures in a chiral magnet, Nature 564, 95 (2018)

    work page 2018

  38. [38]

    Z. Zhu, C. Li, and J. B. Marston, Topology of rotating stratified fluids with and without background shear flow, arXiv e-prints , arXiv:2112.04691 (2021), arXiv:2112.04691. 7

    work page arXiv 2021

  39. [39]

    Lalanne, W

    P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugo- nin, Light interaction with photonic and plasmonic resonances, Laser & Photonics Reviews 12, 1700113 (2018)

    work page 2018

  40. [40]

    Kawabata, T

    K. Kawabata, T. Bessho, and M. Sato, Classification of excep- tional points and non-hermitian topological semimetals, Phys. Rev. Lett. 123, 066405 (2019)

    work page 2019

  41. [41]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-hermitian physics, Phys. Rev. X9, 041015 (2019)

    work page 2019

  42. [42]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Excep- tional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021)

    work page 2021

  43. [43]

    Dennis, Polarization singularities in paraxial vector fields: morphology and statistics, Optics Communications 213, 201 (2002)

    M. Dennis, Polarization singularities in paraxial vector fields: morphology and statistics, Optics Communications 213, 201 (2002). 1 — SUPPLEMENTAL MATERIAL — I. THEORETICAL MODEL In this section, we provide the detailed derivation of the effective Hamiltonian representing leaky Bloch modes in a general photonic lattice of period a along x-axis and exhibit...

    work page 2002

  44. [44]

    It is preferable to work with dimensionless quantities

    Dispersion characteristic of guided modes Assuming that the dispersion characteristic in the vicinity of the Γ1 point is linear with the slope given by the effective group index ng, one may show that the dispersion characteristic ω±1(kx, ky) is given by: ω±1(k) = ωΓ1 + c ng ( |β±1(k)|− 2π a ) ≈ ωΓ1± c ng kx + ca 4πng k2 y (S3) where c is the speed of ligh...

    work page

  45. [45]

    the electric fieldE and magnetic fieldH are in-plan and perpendicular to the propagation vector respectively)

    Polarization vector of the guided modes Polarizations of guided modes are classified into TE and TM modes (i.e. the electric fieldE and magnetic fieldH are in-plan and perpendicular to the propagation vector respectively). Here we suppose that there is only one type of guided modes in the spectral range of interest, thus the setup is with single polarization...

    work page

  46. [46]

    Each of them also couples to the radiative continuum with the coupling strengthγ (see Fig

    Coupling mechanisms: diffractive coupling and radiative coupling Now the implementation of a periodic corrugation will induce a coupling U between|β−1⟩ and|β+1⟩. Each of them also couples to the radiative continuum with the coupling strengthγ (see Fig. S1b). In a general case with a presence of non-radiative losses (if the structures include lossy materia...

    work page

  47. [47]

    Non-Hermitian Hamiltonian: Eigenvalues of Quasi-normal eigenmodes Using the Pauli matrices σ1 = ( 0 1 1 0 ), σ3 = (1 0 0−1 ) , the Hamiltonian (S9) can be rewritten as: H = d0I2 + d1σ1 + d3σ3 (S10) where d0 = q2 y 2ng + i (γ + γnr) , (S11) d1 = U + iγ cos α, (S12) d3 = qx ng . (S13) . The eigenvalues Ω± and corresponding eigenstates|±⟩ of (S10) is given b...

    work page

  48. [48]

    swapping

    Complex eigenvalues of Bloch resonances and band inversion mechanism The explicit form of eigenvalues Ω± are given by: Ω±(q) = q2 y 2ng + i (γ + γnr)± √ q2x n2g + (U + iγ cos α)2 = ˆω± + iγr± (S17) The real part ˆω± of Ω±(q) corresponds to the normalized eigen-frequency. The imaginary counter part γ± corresponds to the losses including both radiative and ...

    work page

  49. [49]

    (S22) At qx = qy = 0, Eq

    Eigenmodes of the effective Hamiltonian The explicit form of eigenstates|±⟩ are given by: |±⟩ =|β+1⟩ + C±|β−1⟩ (S21) with : C± =− qx ng(U− iγ cos α)± √ 1 + [ qx ng(U− iγ cos α) ]2 . (S22) At qx = qy = 0, Eq. (S22) gives us C±|qx=qy=0 =±1, and the eigenmodes is simplifed as: |±qx=qy=0⟩ =|β+1⟩±| β−1⟩ (S23) The π phase between|β+1⟩qx=qy=0 and|β−1⟩qx=qy=0 imp...

    work page

  50. [50]

    Nearfield pattern The nearfield of the guided modes|β±1⟩ is described by plane waves propagating in-plane (x,y) with vertical confinement profilef(z): E ±1 =E0f(z) exp ( iβ±1.r|| ) u±1 =E0f(z) exp ( i2πq.r|| a ) exp ( ±i2πx a ) u±1 (S24) where the propagation wavevector β±1 is from (S2), polarization vector u±1 is from (S7), r|| = ( x, y) is the in-plane coor...

    work page

  51. [51]

    As a consequence, these guided modes once folded radiate to the freespace as plane wave with k as the in-plane component

    Polarization pattern of the farfield in momentum space The Bragg scattering mechanism due to periodic corrugation fold the guided modes|β1⟩ fromβ±1 =k± 2π aux to the same wavevectork. As a consequence, these guided modes once folded radiate to the freespace as plane wave with k as the in-plane component. In the vicinity of the normal emission, these radiat...

    work page

  52. [52]

    (S33) whereC is a closed circulation encircling qx = qy = 0

    Winding number of the polarization vortex From the orientation angle φ±(q), the winding number around qx = qy = 0 of the vector fieldE±(q) is defined as: W± = 1 2π ∮ C dq∇qφ±. (S33) whereC is a closed circulation encircling qx = qy = 0. One may uses directly Eq. (S31) and calculating W± for a arbitrary path C. However, knowing that the only possible singula...

    work page

  53. [53]

    The Fourier plane of the structure located at the focal length of the microscope objective is projected with the ’Fourier’ and ’Focus’ lenses to the entrance of the spectrometer

    Angle-resolved reflectivity measurements All experimental results in this work are obtained from angle-resolved reflectivity measurements using a home-made Fourier setup [see Fig.S3 (a)]. The Fourier plane of the structure located at the focal length of the microscope objective is projected with the ’Fourier’ and ’Focus’ lenses to the entrance of the spectr...

    work page

  54. [54]

    For each kx-resolved measurements at given ky, the measurement is fitted with two Lorentzian functions giving the modes’ wavelength [see blue and red dashed line in Fig

    Tomographic bands reconstruction To perform tomographic bands reconstruction, 50 kx-resolved measurements for ky varying from −kmax to kmax were performed for all the structures. For each kx-resolved measurements at given ky, the measurement is fitted with two Lorentzian functions giving the modes’ wavelength [see blue and red dashed line in Fig. S4 (a)] a...

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  55. [55]

    The tomographic experiment is performed for different polarizations to retrieve polarization-resolved intensities of each mode [see Fig.S5 (a,b)]

    Measurement of the polarization vortex Different polarization elements (polarizers, half-wave plates, quater-wave plates) may be introduced for polarization-resolved measurements. The tomographic experiment is performed for different polarizations to retrieve polarization-resolved intensities of each mode [see Fig.S5 (a,b)]. From these results, we obtain ...

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