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Cited by 3,332Topological Materials and PhenomenaQuantum Mechanics and Non-Hermitian PhysicsAdvanced Condensed Matter Physics

Photonic Floquet topological insulators

Open access
Rechtsman, Mikael C., Zeuner, Julia M., Plotnik, Yonatan, Lumer, Yaakov, Podolsky, Daniel, Dreisow, Felix, Nolte, Stefan, Segev, Mordechai +1 more·2013·Nature
DOI 10.1038/nature12066
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Abstract

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs only on their surfaces. In two dimensions, electrons on the surface of a topological insulator are not scattered despite defects and disorder, providing robustness akin to that of superconductors. Topological insulators are predicted to have wide-ranging applications in fault-tolerant quantum computing and spintronics. Substantial effort has been directed towards realizing topological insulators for electromagnetic waves. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional and therefore exhibit no transport properties. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. But because magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatter-free edge states requires a fundamentally different mechanism—one that is free of magnetic fields. A number of proposals for photonic topological transport have been put forward recently. One suggested temporal modulation of a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states. This is in the spirit of the proposed Floquet topological insulators, in which temporal variations in solid-state systems induce topological edge states. Here we propose and experimentally demonstrate a photonic topological insulator free of external fields and with scatter-free edge transport—a photonic lattice exhibiting topologically protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by a Schrödinger equation where the propagation coordinate (z) acts as ‘time’. Thus the helicity of the waveguides breaks z-reversal symmetry as proposed for Floquet topological insulators. This structure results in one-way edge states that are topologically protected from scattering.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proposes and experimentally realizes a photonic Floquet topological insulator using helical waveguides in a honeycomb lattice, where a fictitious gauge field opens a band gap and creates one-way edge states without any external magnetic field. Light is shown to travel along lattice edges and around corners and defects without backscattering or scattering into the bulk.

Key findings

  • A honeycomb array of helical waveguides exhibits a Floquet band gap at the Dirac points when the helix radius is nonzero; at R=0 the edge states are flat and degenerate, while at R=8 µm they acquire opposite slopes and support clockwise circulation with no counter-propagating edge states.
  • Experiments at 633 nm over 10 cm propagation show light launched near a zig-zag edge moving around a corner onto an armchair edge without backscattering or bulk scattering; a missing waveguide defect is also bypassed without backscattering.
  • The edge-state group velocity depends non-monotonically on helix radius: tight-binding theory predicts a maximum at R=10.3 µm, while experiments suggest the maximum lies between 6 µm and 10 µm; at R=16 µm the group velocity approaches zero and bending losses become large.
  • The authors describe this as the first experimental photonic topological insulator requiring no external field and operating at optical frequencies.

Methods

  • Fused-silica photonic lattices were fabricated by femtosecond direct laser writing, with elliptical waveguides of 11 µm and 3 µm diameters, nearest-neighbor spacing b=15 µm, lattice constant a=26 µm, refractive-index increase of 7×10⁻⁴, and 10 cm propagation length.
  • Propagation was modeled with a paraxial Schrödinger-type equation; transforming to the helical frame produces a spatially uniform circular vector potential, and coupled-mode/tight-binding theory plus Floquet-Bloch analysis gives the band structure and edge-state dispersion.
  • Experiments launched 633 nm beams into edge or corner waveguides of triangular lattices with varying helix radius R, including a lattice with a missing waveguide defect; output facet intensity was imaged and compared with beam-propagation-method simulations.

Limitations

  • Stated in the paper: the experimental maximum group velocity is between 6 µm and 10 µm, whereas tight-binding theory gives 10.3 µm, which the authors attribute to experimental error; a small uncertainty in the coupling constant c(λ) causes simulations to advance slightly farther along the edge than experiments; at R=16 µm bending/radiation losses are large and couple power into free-space scattering modes.
  • Evident from the text: the demonstration is classical, linear, single-wavelength, and limited to specific fabricated honeycomb geometries; the paper does not test nonlinear interactions, entangled photons, or a broad range of disorder types, and topological protection is inferred from absence of backscattering in the reported configurations.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1038/nature12066

Cites

48 of 48 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 13 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    The authors experimentally realized a photonic topological insulator: a photonic lattice that supports topologically protected transport along its edges without any external field.

    “Here, we theoretically propose and experimentally demonstrate the first photonic topological insulator: a photonic lattice exhibiting topologically protected transport on the lattice edges, without the need for any external field.”
    • Also stated in: § Abstract · boosted
      “Here, we present the first photonic topological insulator.Namely, we experimentally demonstrate the topological protection of transport through photonic edge states in a honeycomb photonic lattice system 14,19 -24 (which has been referred to as”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The helical rotation is slow enough that a guided mode remains adiabatically bound to the rotating waveguide.
    • Rests on an assumption: The paraxial Schrödinger equation accurately models light propagation in the waveguide array.
    • Rests on an assumption: Nearest-neighbor coupling alone captures the waveguide interactions in the coupled-mode description.
  • Findingreasoning

    In the helical reference frame, propagation is described by an induced fictitious vector potential mathematically equivalent to a circularly polarized, spatially constant electromagnetic field that breaks chiral symmetry.

    “In the helical reference frame, propagation is described by an induced fictitious vector potential that is mathematically equivalent to a circularly-polarized, spatially-constant electromagnetic field, acting to break the chiral symmetry.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The helical rotation is slow enough that a guided mode remains adiabatically bound to the rotating waveguide.
    • Rests on an assumption: The paraxial Schrödinger equation accurately models light propagation in the waveguide array.
  • Findingown data

    The edge states reside strictly within the bulk band gap of the bulk lattice.

    “These edge states reside strictly within the bulk band gap of the bulk lattice (drawn in black).”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The paraxial Schrödinger equation accurately models light propagation in the waveguide array.
    • Rests on an assumption: Nearest-neighbor coupling alone captures the waveguide interactions in the coupled-mode description.
  • Findingown data

    In a lattice with a missing waveguide defect on the edge, the wavepacket propagates around the defect and continues without backscattering.

    “The wavepacket is launched into the waveguide at the top-right corner (circled with yellow in Fig. 5), and it clearly propagates around the defect and continues forward without backscattering.”
    • Also stated in: § Abstract · plain
      “showing no backscattering despite the presence of the defect (a signature of topological protection)”
    Quote in the paperSubstantial claim
  • Findingown data

    The weak optical intensity in the bulk in simulations comes from overlap of the initial launch beam with bulk modes.

    “The presence of weak optical intensity in the bulk of the lattice arises from overlap of the initial launch beam with the bulk modes of the system.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 16
  • The quote only partly states this claim. 2
  • Classified as probably not a substantial claim. 7

  • May combine two claims.

    Findingown data

    For helical waveguides, the edge states are non-degenerate and propagate only clockwise around the triangular lattice; light at the corner moves along the edge clockwise.

    “When the waveguides are made to spin in a clockwise direction around their axis, the edge states are no longer degenerate and have a preferred direction.In fact, the triangular photonic lattice now has a set of edge states that propagate only clockwise on the circumference of the entire triangle.As a result, light at the corner can no longer remain there, and moves along the edge in the clockwise”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The beam propagates along the top zig-zag edge, hits the corner, and moves down the vertical armchair edge without backscattering or bulk scattering.

    “The beam propagates along the top edge of the array (which is in the zig-zag configuration), hits the corner, and clearly moves down the vertical edge (which is in the armchair configuration).Note that the wavepacket shows no evidence of backscattering or bulk scattering due to its impact with the corner of the lattice.This scattering of the edge state is prevented by topological protection.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the R=0 lattice, launching light at the corner excites bulk and edge states; some light diffracts into the bulk while most remains at the launch corner.

    “while some light has diffracted into the bulk of the lattice, the majority remains at the corner waveguide into which the light was launched.”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingown data

    For R=8μm, the edge states are non-degenerate with opposite slopes; their group velocity is rightward on the top edge and leftward on the bottom edge, giving clockwise circulation, and no counter-clockwise edge states exist.

    “In this case the edge states are no longer degenerate with one another, but now have opposite slopes.Specifically, for this case of waveguides spinning in the clockwise direction, the group velocity on the top edge is directed to the right, and on the bottom edge to the left, corresponding to clockwise circulations.At the same time, there are no edge states whatsoever circulating in the counter-”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The paraxial Schrödinger equation accurately models light propagation in the waveguide array.
    • Rests on an assumption: Nearest-neighbor coupling alone captures the waveguide interactions in the coupled-mode description.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    The simulation wavepacket progressed slightly farther along the edge than the experimental one because of uncertainty in the coupling constant c(λ).

    “Note that the wavepacket in the simulations has progressed slightly farther along the edge than that in the experimental results.This is a result of a small degree of uncertainty in the value of the coupling constant, c(λ).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingcitation

    For straight waveguides (R=0), the honeycomb lattice band structure has Dirac points at conical intersections between the first and second bands and no band gap.

    “Plotted as a function of the transverse Bloch wavevector (k x ,k y ), the conical intersections between the first and second bands are the”
    • Also stated in: § Abstract · plain
      “As a result of these, the band structure is analogous to a metallic phase because there is no band gap.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [28] Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005) source

      Not checked in the free run.

    • Rests on an assumption: The paraxial Schrödinger equation accurately models light propagation in the waveguide array.
    • Rests on an assumption: Nearest-neighbor coupling alone captures the waveguide interactions in the coupled-mode description.

  • May combine two claims.

    Findingown data

    At R=16μm, bending losses are large, causing optical power to leak into scattering modes; the group velocity approaches zero and power remains at the corner.

    “By the point at which R = 16µm, the bending losses are large, leading to a leaking of optical power into scattering modes (which accounts for the large background signal).As shown in Fig. 4(j), the group velocity of the wavepacket approaches zero and therefore the optical power remains at the corner waveguide.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For helical waveguides (R>0), a band gap opens in the Floquet spectrum and the lattice becomes a Floquet topological insulator.

    “When the waveguides are made helical (R>0), a band gap in the Floquet spectrum opens, as shown in Fig. 1(d), and the photonic lattice becomes analogous to an insulator -indeed, to a Floquet topological insulator.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The paraxial Schrödinger equation accurately models light propagation in the waveguide array.
    • Rests on an assumption: Nearest-neighbor coupling alone captures the waveguide interactions in the coupled-mode description.

  • May combine two claims. The quote only partly states this claim.

    Findingown data

    For R=0, the zig-zag edge states form two degenerate sets with flat dispersion between kx=2π/3a and kx=4π/3a, occupying one third of kx space.

    “There are two sets of states residing on the zig-zag edge -one set per edge.Their dispersion curves are entirely flat and completely coincide (i.e., they are degenerate with one another), residing between k x = 2π/3a and k x = 4π/3a, occupying one third of k xspace.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The paraxial Schrödinger equation accurately models light propagation in the waveguide array.
    • Rests on an assumption: Nearest-neighbor coupling alone captures the waveguide interactions in the coupled-mode description.

  • May combine two claims.

    Findingreasoning

    For R=8μm, the wavepacket wraps around the triangle corner and moves along the opposite edge without backscattering at the acute corner.

    “For R=8µm, the wave packet wraps around the corner of the triangle and moves along the opposite edge, as shown in Fig. 4(f).Importantly, the light is not backscattered even when it hits the acute corner, due to the lack of a counter-propagating edge state.This is a key example of topological protection against scattering.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The chirality of the helical waveguides produces one-way edge states that are free of scattering and topologically protected.

    “The chirality of the waveguides results in scatter-free, one-way edge states that are topologically protected from scattering.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Kane and Mele showed that strong spin-orbit coupling can achieve immunity against scattering without an external field.

    “More recently, it was shown by Kane and Mele 5 that in the case of strong spin-orbit coupling, immunity against scattering (topological protection) can also be achieved without the need for an external field: the materials in this class later became known as topological insulators 6 .”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [5] Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009) source

      Not checked in the free run.

    • cites [6] Koch, J., Houck, A. A., Hur, K. L. & Girvin, S. M. Time-reversal-symmetry breaking in circuit-QED-based photon lattices. Phys. Rev. A 82, 043811 (2010) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    In the quantum Hall effect, an external magnetic field breaks time-reversal symmetry, so counter-propagating edge states have different energies and backscattering is prevented.

    “In contrast, in the quantum Hall effect 4 , an external magnetic field is applied to a two-dimensional electron gas, breaking time-reversal symmetry, hence the two counter-propagating edge states have different energies, and thus backscattering is prevented.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [4] Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Graphene has been theorized to exhibit topologically protected edge states under a circularly polarized electric field, but this has not been shown experimentally.

    “Indeed, the two-dimensional material graphene, which has very weak spin-orbit coupling, has been theorized to exhibit topologically protected edge states under the influence of a circularly polarized electric field 8 , but this has not been shown experimentally.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [8] Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907–912 (2011) source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    The photonic lattices have the same geometry as photonic crystal fibers and are likely to exhibit robust topologically protected states.

    “For example, our photonic lattices have the same geometry as photonic crystal fibers, and thus these systems are likely to exhibit robust topologically protected states.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).

    Premisecitation

    It has been conjectured that in Floquet topological insulators spin-orbit coupling is not necessary for topological protection.

    “It has been conjectured that in such materials, called”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [7] Umucalılar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011) source

      Not checked in the free run.

    • cites [8] Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907–912 (2011) source

      Not checked in the free run.

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation

    Raghu and Haldane proposed that gyromagnetic materials could realize a photonic analogue of the quantum Hall effect and topological protection of light.

    “In the domain of optics, Raghu and Handane proposed in 2008 that by using gyromagnetic materials, a photonic analogue of the quantum Hall effect could be realized and topological protection of light could be achieved 9 .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [9] Khanikaev, A. B. et al. Photonic topological insulators. Nature Mater. 12, 233–239 (2012) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Recent works experimentally demonstrated photonic edge states derived by topological arguments, but none are topologically protected against scattering.

    “On the other hand, a number of recent works have experimentally demonstrated the existence of photonic edge states that are derived by topological arguments (namely the bulk-edge correspondence principle) 13 -15 .However, none of these edge states is topologically protected against scattering and should not be confused with the topologically protected edge states in a topological insulator.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [13] Lu, L., Joannopoulos, J. D. & Soljačić, M. Waveguiding at the edge of a three-dimensional photonic crystal. Phys. Rev. Lett. 108, 243901 (2012) source

      Not checked in the free run.

    • cites [14] Malkova, N., Hromada, I., Wang, X., Bryant, G. & Chen, Z. Observation of optical Shockley-like surface states in photonic superlattices. Opt. Lett. 34, 1633–1635 (2009) source

      Not checked in the free run.

    • cites [15] Klitzing, K. v., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    No non-magnetic photonic topological protection has ever been observed experimentally.

    “However, no non-magnetic photonic topological protection has ever been observed experimentally.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisenone given

    Magneto-optic effects at optical frequencies are extremely weak, so the magneto-optic method cannot be carried over to optical frequencies.

    “However, this method of achieving topological protection cannot be carried over to optical frequencies, because magneto-optic effects at optical frequencies are extremely weak.”
    Quote in the paperNot a substantial claim
20 of 20

  • The paper relies on this without support.

    The helical rotation is slow enough that a guided mode remains adiabatically bound to the rotating waveguide.

    “The helical rotation of the waveguides in our honeycomb lattice is sufficiently slow that a guided mode within a waveguide will be adiabatically drawn along with the waveguide as it rotates.”

  • The paper relies on this without support.

    The paraxial Schrödinger equation accurately models light propagation in the waveguide array.

    “The paraxial propagation of light through the photonic lattice is described by the Schrödinger-type equation:”

  • The paper relies on this without support.

    Nearest-neighbor coupling alone captures the waveguide interactions in the coupled-mode description.

    “where the summation is only taken over nearest-neighbor waveguides”
3 of 3

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Optical waveguide axis bending introduces an equivalent gravitational potential for photons.

    “In other words, optical waveguide axis bending introduces an equivalent gravitational potential for photons, which has found several applications, for example in dynamical localization of light [36, 37], control of photon tunneling [38] and realization of synthetic gauge fields for photons [39, 40] (for a recent review see [41]).”

    From Equivalence principle and quantum mechanics: quantum simulation with entangled photons (2018) · cites this paper as [39]

    The passage describing the transformed coordinates (section labeled Abstract; page unavailable) says that helical waveguides produce an equivalent vector potential, supporting the sentence’s example of synthetic gauge fields for photons. It does not describe waveguide bending as introducing an equivalent gravitational potential, which is the specific claim in the quoted text. (LLM, conf=0.82)

    Experimental — accuracy not yet measured

  • When appropriately stacked into three dimensions, the corner modes give rise to chiral one-dimensional modes along edges of the three-dimensional sample.

    “When appropriately stacked into three dimensions, the corner modes give rise to chiral one dimensional modes along edges of the three dimensional sample.”

    From Observation of a phononic quadrupole topological insulator (2018) · cites this paper as [23]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Conflicting launch corner for Figure 5 experiment

    Minor

    The text places the experimental input at the triangle's top-left corner, but the Figure 5 caption places it at the upper-right corner.

1 of 1

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Related

Authors
Rechtsman, M. C., Zeuner, J. M., Plotnik, Y., Lumer, Y., Podolsky, D., Dreisow, F., Nolte, S., Segev, M., & Szameit, A.
Affiliations
  • Technion – Israel Institute of Technology (IL)
  • Friedrich Schiller University Jena (DE)
Corresponding author
  • Rechtsman, Mikael C.
  • Plotnik, Yonatan
ORCID
Venue
Nature
Publisher
Springer Science and Business Media LLC
Published
2013-04
Volume
496
Issue
7444
Pages
196-200
ISSN
  • 0028-0836 (ISSN-L)
  • 1476-4687
DOI
10.1038/nature12066
OpenAlex
W1978426240
Semantic Scholar
54c67b855fa18762e122c93289b197033d6e2669
Cited by
3,332
References
46
Citations by year
2026: 206 · 2025: 243 · 2024: 239 · 2023: 271 · 2022: 307 · 2021: 329 · 2020: 345 · 2019: 270 · 2018: 294 · 2017: 272 · 2016: 213 · 2015: 175 · 2014: 118 · 2013: 48
FWCI
94.97
Type
journal article
Language
en
Open access
Licence
www.springer.com/tdm (tdm)
Locations
  • Nature (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
Funders
Azrieli Foundation
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
Keywords
photonic topological insulators, Floquet topological insulators, helical waveguide, honeycomb lattice, topological protection, disorder scattering, time-reversal symmetry breaking, photonic crystals, photonic lattices, optical waveguides, topological photonics

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