Abstract

We show that 2D photonic lattices consisting of coupled microring resonators can emulate quantum systems driven by periodic Hamiltonians and can thus be used to realize photonic Floquet topological insulators. By transforming a 2D microring lattice into an equivalent array of coupled waveguides with periodic boundary conditions, we explicitly derive the Floquet-Bloch Hamiltonian of the system and determine the winding numbers characterizing the band topology and bulk-edge correspondence of the lattice. By varying the coupling strengths between adjacent resonators, we show that a 2D microring lattice can support both anomalous Floquet insulator edge modes and Chern insulator edge modes over a wide range of coupling parameters.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48(22), 1559 (1982).
    [Crossref]
  2. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011).
    [Crossref]
  3. N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductorquantum wells,” Nat. Phys. 7(6), 490 (2011).
    [Crossref]
  4. A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
    [Crossref]
  5. J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
    [Crossref] [PubMed]
  6. S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
    [Crossref]
  7. W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
    [Crossref] [PubMed]
  8. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
    [Crossref]
  9. 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(19), 195125 (2008).
    [Crossref]
  10. T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010).
    [Crossref]
  11. M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).
  12. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics. 6(11), 782 (2012).
    [Crossref]
  13. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
    [Crossref] [PubMed]
  14. L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nat. Commun. 8, 13756 (2017).
    [Crossref] [PubMed]
  15. G. Q. Liang and Y. D. Chong, “Optical resonator analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
    [Crossref] [PubMed]
  16. M. Pasek and Y. D. Chong, “Network models of photonic floquet topological insulators,” Phys. Rev. B 89(7), 075113 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  20. D. Bachman, A. Tsay, and V. Van, “Negative coupling and coupling phase dispersion in a silicon quadrupole micro-racetrack resonator,” Opt. Express. 23(15), 20089–20095 (2015)
    [Crossref] [PubMed]

2017 (1)

L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nat. Commun. 8, 13756 (2017).
[Crossref] [PubMed]

2016 (1)

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

2015 (3)

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki, “Topological index for periodically driven time-reversal invariant 2D systems,” Phys. Rev. Lett. 114(10), 106806 (2015)
[Crossref] [PubMed]

D. Bachman, A. Tsay, and V. Van, “Negative coupling and coupling phase dispersion in a silicon quadrupole micro-racetrack resonator,” Opt. Express. 23(15), 20089–20095 (2015)
[Crossref] [PubMed]

2014 (2)

M. Pasek and Y. D. Chong, “Network models of photonic floquet topological insulators,” Phys. Rev. B 89(7), 075113 (2014).
[Crossref]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

2013 (5)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

G. Q. Liang and Y. D. Chong, “Optical resonator analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

2012 (1)

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics. 6(11), 782 (2012).
[Crossref]

2011 (3)

A. Tsay and V. Van, “Analytic theory of strongly-coupled microring resonators,” IEEE J. Quantum Electron. 47(7), 997–1005 (2011).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011).
[Crossref]

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductorquantum wells,” Nat. Phys. 7(6), 490 (2011).
[Crossref]

2010 (1)

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010).
[Crossref]

2008 (1)

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(19), 195125 (2008).
[Crossref]

1982 (1)

D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48(22), 1559 (1982).
[Crossref]

Bachman, D.

D. Bachman, A. Tsay, and V. Van, “Negative coupling and coupling phase dispersion in a silicon quadrupole micro-racetrack resonator,” Opt. Express. 23(15), 20089–20095 (2015)
[Crossref] [PubMed]

Barik, S.

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Berg, E.

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010).
[Crossref]

Carpentier, D.

D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki, “Topological index for periodically driven time-reversal invariant 2D systems,” Phys. Rev. Lett. 114(10), 106806 (2015)
[Crossref] [PubMed]

Chan, C. T.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chen, W.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chen, X.-D.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chong, Y. D.

M. Pasek and Y. D. Chong, “Network models of photonic floquet topological insulators,” Phys. Rev. B 89(7), 075113 (2014).
[Crossref]

G. Q. Liang and Y. D. Chong, “Optical resonator analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

DeGottardi, W.

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Delplace, P.

D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki, “Topological index for periodically driven time-reversal invariant 2D systems,” Phys. Rev. Lett. 114(10), 106806 (2015)
[Crossref] [PubMed]

Demler, E.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010).
[Crossref]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011).
[Crossref]

Dong, J.-W.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
[Crossref]

Fan, S.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics. 6(11), 782 (2012).
[Crossref]

Fang, K.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics. 6(11), 782 (2012).
[Crossref]

Fruchart, M.

D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki, “Topological index for periodically driven time-reversal invariant 2D systems,” Phys. Rev. Lett. 114(10), 106806 (2015)
[Crossref] [PubMed]

Furusaki, A.

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(19), 195125 (2008).
[Crossref]

Galitski, V.

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductorquantum wells,” Nat. Phys. 7(6), 490 (2011).
[Crossref]

Gawedzki, K.

D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki, “Topological index for periodically driven time-reversal invariant 2D systems,” Phys. Rev. Lett. 114(10), 106806 (2015)
[Crossref] [PubMed]

Gossard, A. C.

D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48(22), 1559 (1982).
[Crossref]

Hafezi, M.

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011).
[Crossref]

Jiang, S.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

Kitagawa, T.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010).
[Crossref]

Levin, M.

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).

Liang, G. Q.

G. Q. Liang and Y. D. Chong, “Optical resonator analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

Lindner, N. H.

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductorquantum wells,” Nat. Phys. 7(6), 490 (2011).
[Crossref]

Ludwig, A. W. W.

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(19), 195125 (2008).
[Crossref]

Lukin, M. D.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011).
[Crossref]

Lumer, Y.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Macdonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

Maczewsky, L. J.

L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nat. Commun. 8, 13756 (2017).
[Crossref] [PubMed]

Migdall, A.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
[Crossref]

Mittal, S.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
[Crossref]

Miyake, H.

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

Nolte, S.

L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nat. Commun. 8, 13756 (2017).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Pasek, M.

M. Pasek and Y. D. Chong, “Network models of photonic floquet topological insulators,” Phys. Rev. B 89(7), 075113 (2014).
[Crossref]

Plotnik, Y.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Rechtsman, M. C.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Refael, G.

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductorquantum wells,” Nat. Phys. 7(6), 490 (2011).
[Crossref]

Rudner, M.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010).
[Crossref]

Rudner, M. S.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).

Ryu, S.

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(19), 195125 (2008).
[Crossref]

Schnyder, A. P.

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(19), 195125 (2008).
[Crossref]

Segev, M.

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

Stormer, H. L.

D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48(22), 1559 (1982).
[Crossref]

Szameit, A.

L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nat. Commun. 8, 13756 (2017).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Taylor, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
[Crossref]

Taylor, J. M.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011).
[Crossref]

Tsay, A.

D. Bachman, A. Tsay, and V. Van, “Negative coupling and coupling phase dispersion in a silicon quadrupole micro-racetrack resonator,” Opt. Express. 23(15), 20089–20095 (2015)
[Crossref] [PubMed]

A. Tsay and V. Van, “Analytic theory of strongly-coupled microring resonators,” IEEE J. Quantum Electron. 47(7), 997–1005 (2011).
[Crossref]

Tse, W-k

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

Tsui, D. C.

D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48(22), 1559 (1982).
[Crossref]

Van, V.

D. Bachman, A. Tsay, and V. Van, “Negative coupling and coupling phase dispersion in a silicon quadrupole micro-racetrack resonator,” Opt. Express. 23(15), 20089–20095 (2015)
[Crossref] [PubMed]

A. Tsay and V. Van, “Analytic theory of strongly-coupled microring resonators,” IEEE J. Quantum Electron. 47(7), 997–1005 (2011).
[Crossref]

Waks, E.

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Yariv, A.

A. Yariv, Optical electronics, 4th ed. (Saunders, 1991).

Yu, Z.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics. 6(11), 782 (2012).
[Crossref]

Zeuner, J. M.

L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nat. Commun. 8, 13756 (2017).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

Zhou, L.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Zhu, B.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

A. Tsay and V. Van, “Analytic theory of strongly-coupled microring resonators,” IEEE J. Quantum Electron. 47(7), 997–1005 (2011).
[Crossref]

Nat. Commun. (2)

L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nat. Commun. 8, 13756 (2017).
[Crossref] [PubMed]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Nat. Mater. (1)

A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013).
[Crossref]

Nat. Photonics. (2)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics. 6(11), 782 (2012).
[Crossref]

Nat. Phys. (2)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011).
[Crossref]

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductorquantum wells,” Nat. Phys. 7(6), 490 (2011).
[Crossref]

Nature (1)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013).
[Crossref] [PubMed]

New J. Phys. (1)

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Opt. Express. (1)

D. Bachman, A. Tsay, and V. Van, “Negative coupling and coupling phase dispersion in a silicon quadrupole micro-racetrack resonator,” Opt. Express. 23(15), 20089–20095 (2015)
[Crossref] [PubMed]

Phys. Rev. B (3)

M. Pasek and Y. D. Chong, “Network models of photonic floquet topological insulators,” Phys. Rev. B 89(7), 075113 (2014).
[Crossref]

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(19), 195125 (2008).
[Crossref]

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010).
[Crossref]

Phys. Rev. Lett. (4)

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref] [PubMed]

D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48(22), 1559 (1982).
[Crossref]

G. Q. Liang and Y. D. Chong, “Optical resonator analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki, “Topological index for periodically driven time-reversal invariant 2D systems,” Phys. Rev. Lett. 114(10), 106806 (2015)
[Crossref] [PubMed]

Phys. Rev. X (1)

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).

Other (1)

A. Yariv, Optical electronics, 4th ed. (Saunders, 1991).

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic of a 2D square microring lattice with coupling angles θa and θb. (b) Topological transformation of the microring lattice into a coupled waveguide array. Fields of the same colors map between the microring and waveguide lattices. The lines connecting adjacent waveguides represent coupling points between corresponding microrings (for clarity only couplings between the outer waveguides in (b) are shown). (c) Couplings between pairs of waveguides in each of the 4 steps in a driving period.
Fig. 2
Fig. 2 (a) Projected bulk band diagram of an infinite microring lattice with coupling angles θa = 0.45π, θb = 0.05π showing bandgaps I and II. The quasi-energy bands εn are labeled n = 0, 1 and 2. (b) Map of coupling angles θa and θb for which bandgaps I and II are open.
Fig. 3
Fig. 3 Band diagrams of microring lattice strips with 10 unit cells in the y direction, infinite length in the x direction, and coupling angles (a) θa = 0.2π, θb = 0.1π; (b) θa = 0.3π, θb = 0; (c) θa = 0.45π, θb = 0.2π; (d) θa = 0.45π, θb = 0.05π. The winding numbers associated with the open bandgaps I and II and the Chern numbers of the bulk bands are also indicated.
Fig. 4
Fig. 4 Topological phase map of the microring lattice showing the range of coupling angles θa and θb for which CI and AFI edge states are supported in bandgaps I and II (CII means CI edge mode is supported in bandgap I, with similar meanings for AFII and AFIII). The symbol markers correspond to the four examples in Fig. 3.
Fig. 5
Fig. 5 (a) Schematic for evanescent light coupling into and out of a microring lattice. The input and output power coupling efficiencies are assumed to be 99%. Input port 1 is used to excite mode travelling along the top boundary of the lattice; Input port 2 is used to excite mode travelling along the bottom boundary. (b) and (c) Field distributions of chiral edge modes in the microring lattice: (b) AFI edge mode with quasi-energy ε = 0.85π/L in bandgap II; (c) CI edge mode with quasi-energy ε = 0.4π/L in bandgap I.
Fig. 6
Fig. 6 Coupled waveguide array representation of a microring lattice strip with Ny unit cells in the y direction and infinite length in the x direction. The diagrams show the couplings between pairs of waveguides in one column of unit cells in each of the 4 coupling steps in one driving period.

Equations (22)

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i ψ m , n A z = β ψ m , n A κ a ( 1 ) ψ m + 1 , n B κ a ( 2 ) ψ m , n + 1 C κ a ( 3 ) ψ m 1 , n B κ a ( 4 ) ψ m , n 1 C
i z | ψ = H 0 | ψ + H FB | ψ
H FB ( k , z ) = j = 1 4 ( s j H a a ( j ) ( 1 s j ) H a b ( j ) ( 1 s j ) H a b ( j ) s j H b b ( j ) )
H a a ( j ) = ( 0 κ a ( j ) e i b j . k κ a ( j ) e i b j . k 0 )
H b b ( j ) = ( 0 κ b ( j ) e i b j . k κ b ( j ) e i b j . k 0 )
H a b ( j ) = ( κ a ( j ) e i b j . k 0 0 κ b ( j ) e i b j . k )
U ( k , z ) = T exp ( i 0 z H FB ( k , z ) d z )
U ( k , z ) = e i H 1 ( k ) z
U ( k , z ) = e i H 2 ( k ) ( z L / 4 ) e i H 1 ( k ) L / 4
| ψ ( k , z ) = U ( k , z ) e i β z | ψ ( k , 0 )
U ( k , L ) = e i H 4 ( k ) L / 4 e i H 3 ( k ) L / 4 e i H 2 ( k ) L / 4 e i H 1 ( k ) L / 4
C [ P n ] = 1 2 π i d k 2 Tr { P n [ k x P n , k y P n ] }
U ε ( k , z ) = U ( k , z ) V ε ( k , z )
V ε ( k , z ) = e i H eff ( k ) z
U ( k , L ) = e i H eff ( k ) L
W [ U ε ] = 1 8 π 2 0 L d z d k 2 Tr { U ε z U ε [ U ε k x U ε , U ε k y U ε ] }
U ( k , L ) = U 4 U 3 U 2 U 1
U j = diag ( K j , K j , , K j )
K 1 = ( cos θ a i sin θ a 0 0 i sin θ a cos θ a 0 0 0 0 cos θ b i sin θ b 0 0 i sin θ b cos θ b )
K 2 = ( cos θ a 0 i sin θ a 0 0 cos θ b 0 i sin θ b i sin θ a 0 cos θ a 0 0 i sin θ b 0 cos θ b )
K 3 = ( cos θ a i sin θ a e 2 i k x a 0 0 i sin θ a e 2 i k x a cos θ a 0 0 0 0 cos θ b i sin θ b e 2 i k x a 0 0 i sin θ b e 2 i k x a cos θ b )
U 4 = diag ( I , K 4 , K 4 , K 4 , I )

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