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Photonic Floquet topological insulators
Open accessAbstract
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.
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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.
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In the paper
“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”
§ Photonic Floquet Topological Insulators > Abstract
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- 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.
- Also stated in: § Abstract · boosted
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
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- 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These edge states reside strictly within the bulk band gap of the bulk lattice (drawn in black).”
§ Photonic Floquet Topological Insulators > Abstract
plain- Asserted: passes — The authors put it forward as true.
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- 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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)”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“For radius R=12µm, the wavepacket moves along the edge, but with a slower group velocity”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“Taken together, these data show the longitudinal progression of topologically protected modes as they travel along the edge of the lattice.”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“The transverse group velocity of these edge states has a non-trivial dependence on the helical waveguide radius, R. For small R, the group velocity of the edge states increases, but eventually it reaches a maximum and decreases again.”
§ Photonic Floquet Topological Insulators > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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- 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The maximum calculated group velocity is achieved at R=10.3µm.”
- Also stated in: § Abstract · plain
“The maximum occurs at R=10.3µm.”
§ Photonic Floquet Topological Insulators > Abstract
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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- 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.
- Also stated in: § Abstract · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This scattering of the edge state is prevented by topological protection.”
§ Photonic Floquet Topological Insulators > Abstract
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- It may state two separate claims as one. 16
- The quote only partly states this claim. 2
- Probably not a substantial claim. 7
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In the paper
“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”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“while some light has diffracted into the bulk of the lattice, the majority remains at the corner waveguide into which the light was launched.”
§ Photonic Floquet Topological Insulators > Abstract
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“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-”
§ Photonic Floquet Topological Insulators > Abstract
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one. The quote only partly states this 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
What the sources say
- Not checked yetcites [28] Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005) source
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- 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.
- Also stated in: § Abstract · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one. The quote only partly states this 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“The chirality of the waveguides results in scatter-free, one-way edge states that are topologically protected from scattering.”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“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 .”
§ Photonic Floquet Topological Insulators > Abstract
What the sources say
- Not checked yetcites [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 yetcites [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
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
What the sources say
- Not checked yetcites [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
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In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
What the sources say
- Not checked yetcites [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
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“It has been conjectured that in such materials, called”
§ Photonic Floquet Topological Insulators > Abstract
What the sources say
- Not checked yetcites [7] Umucalılar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011) source
- Not checked yetcites [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
hedged- Asserted: fails — The authors may not put it forward as true.
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In the paper
“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 .”
§ Photonic Floquet Topological Insulators > Abstract
What the sources say
- Not checked yetcites [9] Khanikaev, A. B. et al. Photonic topological insulators. Nature Mater. 12, 233–239 (2012) source
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In the paper
“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.”
§ Photonic Floquet Topological Insulators > Abstract
What the sources say
- Not checked yetcites [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 yetcites [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 yetcites [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
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In the paper
“However, no non-magnetic photonic topological protection has ever been observed experimentally.”
§ Photonic Floquet Topological Insulators > Abstract
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In the paper
“However, this method of achieving topological protection cannot be carried over to optical frequencies, because magneto-optic effects at optical frequencies are extremely weak.”
§ Photonic Floquet Topological Insulators > Abstract
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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”
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“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]
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“Many types of topological phases have been observed, for instance, Hall effect [16,17], edge states [18][19][20][21], topological insulators [22,23] and Weyl points [24,25], implying the capability of protecting physical fields.”
From Topological protection of two-photon quantum correlation on a photonic chip (2019) · cites this paper as [22]
In the citing paper
“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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Photonic Floquet Topological Insulators > Abstract
“The excited waveguide is at the top left of the triangle, and the edge state propagates in a clockwise sense, avoiding the defect and propagating through it, eventually hitting the next corner of the triangle.”
§ Photonic Floquet Topological Insulators > Abstract
“A light beam of λ=633nm is launched into the single waveguide at the upper-right corner (surrounded by a yellow circle).”
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Kirsch, M. S., Zhang, Y., Kremer, M., Maczewsky, L. J., Ivanov, S. K., Kartashov, Y. V., Torner, L., Bauer, D., Szameit, A., & Heinrich, M. (2021) · Nature Physics · doi:10.1038/s41567-021-01275-3 · Cited by 251
Self-citationTunable Topological Phononic Crystals
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Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology
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Metadata
- 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
- Published
- 2013-04
- Volume
- 496
- Issue
- 7444
- Pages
- 196-200
- ISSN
- 0028-0836 (ISSN-L)
- 1476-4687
- 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
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Nature (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Funders
- Azrieli Foundation
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › 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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-06, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-06 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
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