Abstract

We show that Floquet topological insulating states can exist in two-dimensional photonic crystals made of time-variant optical materials. By arranging the modulating phases, it facilitates effective gauge fields that give rise to topological effects. The band structures demonstrate the existence of topologically non-trivial bandgaps, thereby leading to back-scattering immune unidirectional edge states owing to bulk-edge correspondence. With these first-principle numerical results, we then verify the topological order for every Floquet band via Wilson loop approach. In the final paragraph, the possible experimental implementation for Floquet topological photonics is also discussed.

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  1. A. K. Kamal, “A Parametric Device as a Nonreciprocal Element,” Proc. IRE 48(8), 1424–1430 (1960).
    [Crossref]
  2. M. R. Currie and R. W. Gould, “Coupled-Cavity Traveling-Wave Parametric Amplifiers: Part I - Analysis,” Proc. IRE 48(12), 1960–1973 (1960).
    [Crossref]
  3. E. S. Cassedy and A. A. Oliner, “Dispersion relations in time-space periodic media: Part I - Stable interactions,” Proc. IEEE 51(10), 1342–1359 (1963).
    [Crossref]
  4. E. S. Cassedy, “Dispersion relations in time-space periodic media part II - Unstable interactions,” Proc. IEEE 55(7), 1154–1168 (1967).
    [Crossref]
  5. A. Hessel and A. A. Oliner, “Wave Propagation in a Medium with a Progressive Sinusoidal Disturbance,” IEEE Trans. Microwave Theory Techn. 9(4), 337–343 (1961).
    [Crossref]
  6. D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14(2), 183–194 (1966).
    [Crossref]
  7. A. Fettweis, “Steady-State Analysis of Circuits Containing a Periodically-Operated Switch,” IRE Trans. Circuit Theory 6(3), 252–260 (1959).
    [Crossref]
  8. M. L. Liou, “Exact Analysis of Linear Circuits Containing Periodically Operated Switches with Applications,” IEEE Trans. Circuit Theory 19(2), 146–154 (1972).
    [Crossref]
  9. T. Strom and S. Signell, “Analysis of periodically switched linear circuits,” IEEE Trans. Circuits Syst. 24(10), 531–541 (1977).
    [Crossref]
  10. Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
    [Crossref]
  11. K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm Effect Based on Dynamic Modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
    [Crossref]
  12. D. L. Sounas and A. Alu, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
    [Crossref]
  13. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6(11), 782–787 (2012).
    [Crossref]
  14. E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband Nonreciprocal Amplification in Luminal Metamaterials,” Phys. Rev. Lett. 123(20), 206101 (2019).
    [Crossref]
  15. J. Cayssol, B. Dra, F. Simon, and R. Moessner, “Floquet topological insulators,” Phys. Status Solidi RRL 7(1-2), 101–108 (2013).
    [Crossref]
  16. R. Fleury, A. Khanikaev, and A. Alù, “Floquet topological insulators for sound,” Nat. Commun. 7(1), 11744 (2016).
    [Crossref]
  17. M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
    [Crossref]
  18. Y. T. Wang and S. Zhang, “Elastic Spin-Hall effect in Mechanical Graphene,” New J. Phys. 18(11), 113014 (2016).
    [Crossref]
  19. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
    [Crossref]
  20. Y. T. Wang, P. G. Luan, and S. Zhang, “Coriolis force induced topological order for classical mechanical vibrations,” New. J. Phys. 17(7), 073031 (2015).
    [Crossref]
  21. M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
    [Crossref]
  22. H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, “Quantum anomalous Hall effect and related topological electronic states,” Adv. Phys. 64(3), 227–282 (2015).
    [Crossref]
  23. H. X. Wang, G. Y. Guo, and J. H. Jiang, “Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology,” New J. Phys. 21(9), 093029 (2019).
    [Crossref]
  24. K. G. Wilson, “Confinement of quarks,” Phys. Rev. D 10(8), 2445–2459 (1974).
    [Crossref]
  25. R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, “Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection,” Phys. Rev. B 84(7), 075119 (2011).
    [Crossref]
  26. A. Alexandradinata, X. Dai, and B. A. Bernevig, “Wilson-loop characterization of inversion-symmetric topological insulators,” Phys. Rev. B 89(15), 155114 (2014).
    [Crossref]
  27. X. Wang, C. Li, D. Song, and R. Dean, “A Nonlinear Circuit Analysis Technique for Time-Variant Inductor Systems,” Sensors 19(10), 2321 (2019).
    [Crossref]
  28. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996).
    [Crossref]

2019 (3)

E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband Nonreciprocal Amplification in Luminal Metamaterials,” Phys. Rev. Lett. 123(20), 206101 (2019).
[Crossref]

H. X. Wang, G. Y. Guo, and J. H. Jiang, “Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology,” New J. Phys. 21(9), 093029 (2019).
[Crossref]

X. Wang, C. Li, D. Song, and R. Dean, “A Nonlinear Circuit Analysis Technique for Time-Variant Inductor Systems,” Sensors 19(10), 2321 (2019).
[Crossref]

2017 (1)

D. L. Sounas and A. Alu, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
[Crossref]

2016 (2)

R. Fleury, A. Khanikaev, and A. Alù, “Floquet topological insulators for sound,” Nat. Commun. 7(1), 11744 (2016).
[Crossref]

Y. T. Wang and S. Zhang, “Elastic Spin-Hall effect in Mechanical Graphene,” New J. Phys. 18(11), 113014 (2016).
[Crossref]

2015 (4)

Y. T. Wang, P. G. Luan, and S. Zhang, “Coriolis force induced topological order for classical mechanical vibrations,” New. J. Phys. 17(7), 073031 (2015).
[Crossref]

M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
[Crossref]

H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, “Quantum anomalous Hall effect and related topological electronic states,” Adv. Phys. 64(3), 227–282 (2015).
[Crossref]

Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
[Crossref]

2014 (1)

A. Alexandradinata, X. Dai, and B. A. Bernevig, “Wilson-loop characterization of inversion-symmetric topological insulators,” Phys. Rev. B 89(15), 155114 (2014).
[Crossref]

2013 (2)

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref]

J. Cayssol, B. Dra, F. Simon, and R. Moessner, “Floquet topological insulators,” Phys. Status Solidi RRL 7(1-2), 101–108 (2013).
[Crossref]

2012 (2)

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm Effect Based on Dynamic Modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[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), 782–787 (2012).
[Crossref]

2011 (1)

R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, “Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection,” Phys. Rev. B 84(7), 075119 (2011).
[Crossref]

2008 (1)

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996).
[Crossref]

1977 (1)

T. Strom and S. Signell, “Analysis of periodically switched linear circuits,” IEEE Trans. Circuits Syst. 24(10), 531–541 (1977).
[Crossref]

1974 (1)

K. G. Wilson, “Confinement of quarks,” Phys. Rev. D 10(8), 2445–2459 (1974).
[Crossref]

1972 (1)

M. L. Liou, “Exact Analysis of Linear Circuits Containing Periodically Operated Switches with Applications,” IEEE Trans. Circuit Theory 19(2), 146–154 (1972).
[Crossref]

1967 (1)

E. S. Cassedy, “Dispersion relations in time-space periodic media part II - Unstable interactions,” Proc. IEEE 55(7), 1154–1168 (1967).
[Crossref]

1966 (1)

D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14(2), 183–194 (1966).
[Crossref]

1963 (1)

E. S. Cassedy and A. A. Oliner, “Dispersion relations in time-space periodic media: Part I - Stable interactions,” Proc. IEEE 51(10), 1342–1359 (1963).
[Crossref]

1961 (1)

A. Hessel and A. A. Oliner, “Wave Propagation in a Medium with a Progressive Sinusoidal Disturbance,” IEEE Trans. Microwave Theory Techn. 9(4), 337–343 (1961).
[Crossref]

1960 (2)

A. K. Kamal, “A Parametric Device as a Nonreciprocal Element,” Proc. IRE 48(8), 1424–1430 (1960).
[Crossref]

M. R. Currie and R. W. Gould, “Coupled-Cavity Traveling-Wave Parametric Amplifiers: Part I - Analysis,” Proc. IRE 48(12), 1960–1973 (1960).
[Crossref]

1959 (1)

A. Fettweis, “Steady-State Analysis of Circuits Containing a Periodically-Operated Switch,” IRE Trans. Circuit Theory 6(3), 252–260 (1959).
[Crossref]

Alexandradinata, A.

A. Alexandradinata, X. Dai, and B. A. Bernevig, “Wilson-loop characterization of inversion-symmetric topological insulators,” Phys. Rev. B 89(15), 155114 (2014).
[Crossref]

Alu, A.

D. L. Sounas and A. Alu, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
[Crossref]

Alù, A.

R. Fleury, A. Khanikaev, and A. Alù, “Floquet topological insulators for sound,” Nat. Commun. 7(1), 11744 (2016).
[Crossref]

Bernevig, A.

R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, “Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection,” Phys. Rev. B 84(7), 075119 (2011).
[Crossref]

Bernevig, B. A.

A. Alexandradinata, X. Dai, and B. A. Bernevig, “Wilson-loop characterization of inversion-symmetric topological insulators,” Phys. Rev. B 89(15), 155114 (2014).
[Crossref]

Cassedy, E. S.

E. S. Cassedy, “Dispersion relations in time-space periodic media part II - Unstable interactions,” Proc. IEEE 55(7), 1154–1168 (1967).
[Crossref]

E. S. Cassedy and A. A. Oliner, “Dispersion relations in time-space periodic media: Part I - Stable interactions,” Proc. IEEE 51(10), 1342–1359 (1963).
[Crossref]

Cayssol, J.

J. Cayssol, B. Dra, F. Simon, and R. Moessner, “Floquet topological insulators,” Phys. Status Solidi RRL 7(1-2), 101–108 (2013).
[Crossref]

Chan, C. T.

M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
[Crossref]

Chen, W. J.

M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
[Crossref]

Chen, Y.-F.

Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
[Crossref]

Chen, Z.-G.

Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
[Crossref]

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Currie, M. R.

M. R. Currie and R. W. Gould, “Coupled-Cavity Traveling-Wave Parametric Amplifiers: Part I - Analysis,” Proc. IRE 48(12), 1960–1973 (1960).
[Crossref]

Dai, X.

H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, “Quantum anomalous Hall effect and related topological electronic states,” Adv. Phys. 64(3), 227–282 (2015).
[Crossref]

A. Alexandradinata, X. Dai, and B. A. Bernevig, “Wilson-loop characterization of inversion-symmetric topological insulators,” Phys. Rev. B 89(15), 155114 (2014).
[Crossref]

R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, “Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection,” Phys. Rev. B 84(7), 075119 (2011).
[Crossref]

Dean, R.

X. Wang, C. Li, D. Song, and R. Dean, “A Nonlinear Circuit Analysis Technique for Time-Variant Inductor Systems,” Sensors 19(10), 2321 (2019).
[Crossref]

Dra, B.

J. Cayssol, B. Dra, F. Simon, and R. Moessner, “Floquet topological insulators,” Phys. Status Solidi RRL 7(1-2), 101–108 (2013).
[Crossref]

Dreisow, F.

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (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), 782–787 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm Effect Based on Dynamic Modulation,” Phys. Rev. Lett. 108(15), 153901 (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), 782–787 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm Effect Based on Dynamic Modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref]

Fang, Z.

H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, “Quantum anomalous Hall effect and related topological electronic states,” Adv. Phys. 64(3), 227–282 (2015).
[Crossref]

R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, “Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection,” Phys. Rev. B 84(7), 075119 (2011).
[Crossref]

Fettweis, A.

A. Fettweis, “Steady-State Analysis of Circuits Containing a Periodically-Operated Switch,” IRE Trans. Circuit Theory 6(3), 252–260 (1959).
[Crossref]

Fleury, R.

R. Fleury, A. Khanikaev, and A. Alù, “Floquet topological insulators for sound,” Nat. Commun. 7(1), 11744 (2016).
[Crossref]

Galiffi, E.

E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband Nonreciprocal Amplification in Luminal Metamaterials,” Phys. Rev. Lett. 123(20), 206101 (2019).
[Crossref]

Gould, R. W.

M. R. Currie and R. W. Gould, “Coupled-Cavity Traveling-Wave Parametric Amplifiers: Part I - Analysis,” Proc. IRE 48(12), 1960–1973 (1960).
[Crossref]

Guo, G. Y.

H. X. Wang, G. Y. Guo, and J. H. Jiang, “Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology,” New J. Phys. 21(9), 093029 (2019).
[Crossref]

He, W. Y.

M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
[Crossref]

Hessel, A.

A. Hessel and A. A. Oliner, “Wave Propagation in a Medium with a Progressive Sinusoidal Disturbance,” IEEE Trans. Microwave Theory Techn. 9(4), 337–343 (1961).
[Crossref]

Holberg, D. E.

D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14(2), 183–194 (1966).
[Crossref]

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996).
[Crossref]

Hu, X.

H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, “Quantum anomalous Hall effect and related topological electronic states,” Adv. Phys. 64(3), 227–282 (2015).
[Crossref]

Huidobro, P. A.

E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband Nonreciprocal Amplification in Luminal Metamaterials,” Phys. Rev. Lett. 123(20), 206101 (2019).
[Crossref]

Jiang, J. H.

H. X. Wang, G. Y. Guo, and J. H. Jiang, “Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology,” New J. Phys. 21(9), 093029 (2019).
[Crossref]

Joannopoulos, J. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Kamal, A. K.

A. K. Kamal, “A Parametric Device as a Nonreciprocal Element,” Proc. IRE 48(8), 1424–1430 (1960).
[Crossref]

Khanikaev, A.

R. Fleury, A. Khanikaev, and A. Alù, “Floquet topological insulators for sound,” Nat. Commun. 7(1), 11744 (2016).
[Crossref]

Kunz, K. S.

D. E. Holberg and K. S. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14(2), 183–194 (1966).
[Crossref]

Li, C.

X. Wang, C. Li, D. Song, and R. Dean, “A Nonlinear Circuit Analysis Technique for Time-Variant Inductor Systems,” Sensors 19(10), 2321 (2019).
[Crossref]

Liou, M. L.

M. L. Liou, “Exact Analysis of Linear Circuits Containing Periodically Operated Switches with Applications,” IEEE Trans. Circuit Theory 19(2), 146–154 (1972).
[Crossref]

Liu, X.-P.

Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
[Crossref]

Lu, M.-H.

Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
[Crossref]

Luan, P. G.

Y. T. Wang, P. G. Luan, and S. Zhang, “Coriolis force induced topological order for classical mechanical vibrations,” New. J. Phys. 17(7), 073031 (2015).
[Crossref]

Lumer, Y.

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref]

Moessner, R.

J. Cayssol, B. Dra, F. Simon, and R. Moessner, “Floquet topological insulators,” Phys. Status Solidi RRL 7(1-2), 101–108 (2013).
[Crossref]

Ni, X.

Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
[Crossref]

Nolte, S.

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref]

Oliner, A. A.

E. S. Cassedy and A. A. Oliner, “Dispersion relations in time-space periodic media: Part I - Stable interactions,” Proc. IEEE 51(10), 1342–1359 (1963).
[Crossref]

A. Hessel and A. A. Oliner, “Wave Propagation in a Medium with a Progressive Sinusoidal Disturbance,” IEEE Trans. Microwave Theory Techn. 9(4), 337–343 (1961).
[Crossref]

Pendry, J. B.

E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband Nonreciprocal Amplification in Luminal Metamaterials,” Phys. Rev. Lett. 123(20), 206101 (2019).
[Crossref]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996).
[Crossref]

Plotnik, Y.

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref]

Podolsky, D.

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref]

Qi, X. L.

R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, “Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection,” Phys. Rev. B 84(7), 075119 (2011).
[Crossref]

Rechtsman, M.

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref]

Segev, M.

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref]

Signell, S.

T. Strom and S. Signell, “Analysis of periodically switched linear circuits,” IEEE Trans. Circuits Syst. 24(10), 531–541 (1977).
[Crossref]

Simon, F.

J. Cayssol, B. Dra, F. Simon, and R. Moessner, “Floquet topological insulators,” Phys. Status Solidi RRL 7(1-2), 101–108 (2013).
[Crossref]

Soljacic, M.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Song, D.

X. Wang, C. Li, D. Song, and R. Dean, “A Nonlinear Circuit Analysis Technique for Time-Variant Inductor Systems,” Sensors 19(10), 2321 (2019).
[Crossref]

Sounas, D. L.

D. L. Sounas and A. Alu, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
[Crossref]

Stewart, W. J.

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Nature (1)

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K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm Effect Based on Dynamic Modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
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Q. Wang, Y. Yang, X. Ni, Y.-L. Xu, X.-C. Sun, Z.-G. Chen, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Acoustic asymmetric transmission based on time-dependent dynamical scattering,” Sci. Rep. 5(1), 10880 (2015).
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X. Wang, C. Li, D. Song, and R. Dean, “A Nonlinear Circuit Analysis Technique for Time-Variant Inductor Systems,” Sensors 19(10), 2321 (2019).
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Figures (4)

Fig. 1.
Fig. 1. (a) The schematic diagram of Floquet topological photonic crystals. The yellow, blue, and green represent the phase modulations equal to 0, 2π/3, and 4π/3, respectively. The grey area illustrates an unmodulated material such as dielectrics. (b) Band structures of time-invariant (blue dash lines) and time-variant (red solid lines) cases. Three non-trivial band gaps arise when the modulation applies. The parameters are ${\varepsilon _m} = 2.25$ , δ = 0.15, and Ω = 0.1.
Fig. 2.
Fig. 2. Berry phase evolutions demonstrates the winding number for each band. For band 1(4), the evolution circulates once and then accumulates an additional negative(positive) $\pi $ shift, leading to Chern number of a minus(plus) one. In contrast, Chern numbers of band 2 and 3 are both zero as there exist no circulation within Brillouin zone.
Fig. 3.
Fig. 3. (a) Projected band structure with truncation along x axis shows topological edge states (colored in red and green) within non-trivial bandgaps. (b) The eigen-solutions of three non-trivial edge states demonstrate the field confinement near the top and bottom PEC boundaries. At frequency equal to 3.0924, the field profiles illustrate unidirectional edge modes (yellow arrows indicate the direction of propagation) with topological protection in (c) and (d). The inset illustrates the top view of the structure and the star sign labels where the source is located.
Fig. 4.
Fig. 4. (a) The possible realization for Floquet topological photonic crystal consisting of 48 active metallic wires. The phase modulation arrangement remains the same as Fig. 2. (b) Band structures with temporal modulations show three topologically non-trivial band gaps similar to Fig. 2. The parameters are δ = 0.15, $\omega _p^2({{L_0} + {L_1}} )= 25$ , and Ω = 0.1. (c) Berry phase evolutions for the first and fourth bands present that the winding numbers are equal to ∓1.

Equations (7)

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ε j = ε m [ 1 + δ cos ( Ω t + ϕ j ) ] ,
1 μ E z , n + ( ω + n Ω ) 2 c 2 [ ε m E z , n + δ ε m 2 ( e i ϕ j E z , n 1 + e i ϕ j E z , n + 1 ) ] = 0.
C n = 1 2 π π π d k 1 k 1 [ π π d k 2 A n , k ] = 1 2 π π π d θ n , k 1 ,
W ( k 1 ) = j = 1 N E z , n ( k 1 , k 2 , j ) | E z , n ( k 1 , k 2 , j ) E z , n ( k 1 , k 2 , j + 1 ) E z , n ( k 1 , k 2 , j + 1 ) ,
θ n , k 1 Im log ( w n , k 1 ) .
d ( L m J z ) d t = ε 0 α E z ,
1 μ E z ε c 2 2 t 2 E z μ 0 t J z = 0 .

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