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Photonic Floquet topological insulators

Abstract

Topological insulators are a new phase of matter1, with the striking property that conduction of electrons occurs only on their surfaces1,2,3. 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 waves4,5,6,7,8,9,10,11,12,13. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional and therefore exhibit no transport properties11,12,14. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect15, by placing a gyromagnetic photonic crystal in an external magnetic field5. 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 recently6,7,8,9,10. One suggested temporal modulation of a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states10. This is in the spirit of the proposed Floquet topological insulators16,17,18,19, 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 waveguides20 arranged in a graphene-like honeycomb lattice21,22,23,24,25,26. Paraxial diffraction of light is described by a Schrödinger equation where the propagation coordinate (z) acts as ‘time’27. 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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Figure 1: Geometry and band structure of honeycomb photonic Floquet topological insulator lattice.
Figure 2: Dispersion curves of the edge states, highlighting the unique dispersion properties of the topologically protected edge states for helical waveguides in the honeycomb lattice.
Figure 3: Light emerging from the output facet of the waveguide array as the input beam is moved rightwards, along the top edge of the waveguide array.
Figure 4: Experiments highlighting light circulation along the edges of a triangular-shaped lattice of helical waveguides arranged in a honeycomb geometry.
Figure 5: Experiments and simulations showing topological protection in the presence of a defect.

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Acknowledgements

M.C.R. is grateful to the Azrieli Foundation for the Azrieli fellowship while at the Technion. M.S. acknowledges the support of the Israel Science Foundation, the USA-Israel Binational Science Foundation, and an Advanced Grant from the European Research Council. A.S. thanks the German Ministry of Education and Research (Center for Innovation Competence program, grant 03Z1HN31) and the Thuringian Ministry for Education, Science and Culture (Research group Spacetime, grant 11027-514) for support. The authors thank S. Raghu and T. Pereg-Barnea for discussions.

Author information

Authors and Affiliations

Authors

Contributions

The idea was conceived by Y.P. and M.C.R. The theory was investigated by M.C.R. and Y.P. The fabrication was carried out by J.M.Z. The experiments were carried out by M.C.R., Y.P. and J.M.Z. All authors contributed considerably.

Corresponding authors

Correspondence to Mikael C. Rechtsman or Yonatan Plotnik.

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The authors declare no competing financial interests.

Supplementary information

Scatter-free propagation in a rectangular array

This animated video shows the simulation of the propagation of an optical wavepacket through a honeycomb photonic lattice of helical waveguides, from z=0cm to z=10cm, with helix radius R=8μm. The waveguide lattice is rectangular in shape, and the initial beam is incident on the top edge (as in the experimental results of Fig. 3). During propagation, the beam hits the corner of the array, does not backscatter, and propagates downwards along the armchair edge. This constitutes a demonstration of topological protection. (MOV 2287 kb)

Corner excitation in triangular-shaped array with straight waveguides

This animated video shows the simulation of optical propagation through a honeycomb lattice, where the whole lattice is triangularly-shaped (see Fig. 4(a)), and the initial beam is incident upon the top-left waveguide. Here, the waveguides are straight (R=0), implying the edge states have zero group velocity (see Fig. 2(a) and 2(c)), and therefore the wavefunction remains trapped at the corner (besides some expected, small bulk excitation). (MOV 1681 kb)

Corner excitation in triangular-shaped array with rotating waveguides

This animated video shows the simulation of optical propagation through a honeycomb lattice, where the whole lattice is triangularly-shaped (see Fig. 4(a)), and the initial beam is incident upon the top-left waveguide. Here, the waveguides are helical (R=10μm), implying the edge states have non-zero group velocity in the clockwise direction (see Fig. 2(b) and 2(c)), and therefore the wavefunction moves along the top edge in a clockwise sense. The wavefunction wraps around the top-right corner of the triangle without backscattering, implying topological protection. (MOV 1665 kb)

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Rechtsman, M., Zeuner, J., Plotnik, Y. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013). https://doi.org/10.1038/nature12066

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