Abstract

Utilizing the robust transport properties of the topological photonic crystal interface, we experimentally realize two-dimensional topological photonic crystal cavities, where discrete whispering gallery modes can propagate unidirectionally along the cavity circumference. Different from traditional cavities, these topological whispering galley modes are insensitive to cavity shapes. Our microwave demonstration has a good agreement with numerical simulations. Using pure dielectrics, by scaling down to the optical wavelength, an optical directional coupler based on the same topological photonic crystal scheme is also proposed. We here show that topological photonics can provide more novel designs for optical devices.

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

Full Article  |  PDF Article
OSA Recommended Articles
Zak phase induced multiband waveguide by two-dimensional photonic crystals

Yuting Yang, Tao Xu, Yun Fei Xu, and Zhi Hong Hang
Opt. Lett. 42(16) 3085-3088 (2017)

Realization of a complementary medium using dielectric photonic crystals

Tao Xu, Anan Fang, Ziyuan Jia, Liyu Ji, and Zhi Hong Hang
Opt. Lett. 42(23) 4909-4912 (2017)

Efficient slow light coupling into photonic crystals

C. Martijn de Sterke, J. Walker, Kokou B. Dossou, and Lindsay C. Botten
Opt. Express 15(17) 10984-10990 (2007)

References

  • View by:
  • |
  • |
  • |

  1. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
    [Crossref]
  2. A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
    [Crossref]
  3. K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
    [Crossref]
  4. F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
    [Crossref] [PubMed]
  5. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
    [Crossref] [PubMed]
  6. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
    [Crossref] [PubMed]
  7. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
    [Crossref] [PubMed]
  8. K. Sakoda, Optical properties of photonic crystals (Springer Science & Business Media, 2004).
  9. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton university press, 2011).
  10. C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
    [Crossref] [PubMed]
  11. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
    [Crossref] [PubMed]
  12. A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
    [Crossref] [PubMed]
  13. 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(1), 5782 (2014).
    [Crossref] [PubMed]
  14. L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
    [Crossref] [PubMed]
  15. T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
    [Crossref]
  16. C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
    [Crossref]
  17. Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
    [Crossref] [PubMed]
  18. S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
    [Crossref] [PubMed]
  19. S. Yang, Y. Wang, and H. Sun, “Advances and prospects for whispering gallery mode microcavities,” Adv. Opt. Materials 3(9), 1136–1162 (2015).
    [Crossref]
  20. M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
    [Crossref] [PubMed]
  21. R. Chen, V. D. Ta, and H. Sun, “Bending-induced bidirectional tuning of whispering gallery mode lasing from flexible polymer fibers,” ACS Photonics 1(1), 11–16 (2014).
    [Crossref]
  22. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
    [Crossref] [PubMed]
  23. S. Blair and Y. Chen, “Resonant-enhanced evanescent-wave fluorescence biosensing with cylindrical optical cavities,” Appl. Opt. 40(4), 570–582 (2001).
    [Crossref] [PubMed]
  24. C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
    [Crossref]
  25. I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31(9), 1319–1321 (2006).
    [Crossref] [PubMed]
  26. J. D. Suter, W. Lee, D. J. Howard, E. Hoppmann, I. M. White, and X. Fan, “Demonstration of the coupling of optofluidic ring resonator lasers with liquid waveguides,” Opt. Lett. 35(17), 2997–2999 (2010).
    [Crossref] [PubMed]
  27. B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
    [Crossref] [PubMed]
  28. M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).
  29. Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
    [Crossref]
  30. G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
    [Crossref]
  31. Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci. Rep. 8(1), 1588 (2018).
    [Crossref] [PubMed]
  32. L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
    [Crossref]

2018 (3)

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci. Rep. 8(1), 1588 (2018).
[Crossref] [PubMed]

2017 (6)

L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
[Crossref]

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
[Crossref]

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
[Crossref] [PubMed]

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[Crossref]

2016 (2)

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

2015 (3)

S. Yang, Y. Wang, and H. Sun, “Advances and prospects for whispering gallery mode microcavities,” Adv. Opt. Materials 3(9), 1136–1162 (2015).
[Crossref]

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

2014 (3)

R. Chen, V. D. Ta, and H. Sun, “Bending-induced bidirectional tuning of whispering gallery mode lasing from flexible polymer fibers,” ACS Photonics 1(1), 11–16 (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(1), 5782 (2014).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

2013 (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), 233–239 (2013).
[Crossref] [PubMed]

2010 (1)

2009 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

2008 (2)

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[Crossref] [PubMed]

2006 (2)

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[Crossref] [PubMed]

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31(9), 1319–1321 (2006).
[Crossref] [PubMed]

2005 (1)

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[Crossref] [PubMed]

2003 (1)

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

2001 (1)

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[Crossref] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[Crossref] [PubMed]

1980 (1)

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Arnold, S.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[Crossref] [PubMed]

Bahari, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

Bandres, M. A.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

Bernevig, B. A.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[Crossref] [PubMed]

Berthelot, T.

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
[Crossref] [PubMed]

Blair, S.

Chan, C. T.

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[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(1), 5782 (2014).
[Crossref] [PubMed]

Chao, C.-Y.

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

Chen, R.

R. Chen, V. D. Ta, and H. Sun, “Bending-induced bidirectional tuning of whispering gallery mode lasing from flexible polymer fibers,” ACS Photonics 1(1), 11–16 (2014).
[Crossref]

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(1), 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(1), 5782 (2014).
[Crossref] [PubMed]

Chen, Y.

Chen, Y.-B.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Chen, Y.-F.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Christodoulides, D. N.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

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(1), 5782 (2014).
[Crossref] [PubMed]

Dorda, G.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

El Amili, A.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

Fainman, Y.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

Fan, X.

Fink, M.

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
[Crossref] [PubMed]

Fleury, R.

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
[Crossref] [PubMed]

Foreman, M. R.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

Ge, H.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Giannini, V.

G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
[Crossref]

Guo, L. J.

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

Haldane, F. D.

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Hang, Z. H.

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci. Rep. 8(1), 1588 (2018).
[Crossref] [PubMed]

Harari, G.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

He, C.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Hong Hang, Z.

L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
[Crossref]

Hoppmann, E.

Howard, D. J.

Hu, X.

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

Hughes, T. L.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[Crossref] [PubMed]

Huidobro, P. A.

G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
[Crossref]

Jiang, H.

Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci. Rep. 8(1), 1588 (2018).
[Crossref] [PubMed]

Jiang, J.-H.

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

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(1), 5782 (2014).
[Crossref] [PubMed]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[Crossref] [PubMed]

Kane, C. L.

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[Crossref] [PubMed]

Kanté, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[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), 233–239 (2013).
[Crossref] [PubMed]

Khajavikhan, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

Khanikaev, A. B.

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[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), 233–239 (2013).
[Crossref] [PubMed]

Klitzing, K. V.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Lee, W.

Lemoult, F.

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
[Crossref] [PubMed]

Lerosey, G.

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
[Crossref] [PubMed]

Liu, H.

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

Liu, X.-P.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Liu, Z.

L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
[Crossref]

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Lu, M.-H.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Ma, T.

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

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), 233–239 (2013).
[Crossref] [PubMed]

Mele, E. J.

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[Crossref] [PubMed]

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), 233–239 (2013).
[Crossref] [PubMed]

Ndao, A.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

Ni, X.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Oveys, H.

Parto, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

Pepper, M.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Qiu, C.

L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
[Crossref]

Raghu, S.

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Ren, J.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

Segev, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

Shvets, G.

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[Crossref]

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[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), 233–239 (2013).
[Crossref] [PubMed]

Siroki, G.

G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
[Crossref]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Sun, H.

S. Yang, Y. Wang, and H. Sun, “Advances and prospects for whispering gallery mode microcavities,” Adv. Opt. Materials 3(9), 1136–1162 (2015).
[Crossref]

R. Chen, V. D. Ta, and H. Sun, “Bending-induced bidirectional tuning of whispering gallery mode lasing from flexible polymer fibers,” ACS Photonics 1(1), 11–16 (2014).
[Crossref]

Sun, X.-C.

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Suter, J. D.

Swaim, J. D.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

Ta, V. D.

R. Chen, V. D. Ta, and H. Sun, “Bending-induced bidirectional tuning of whispering gallery mode lasing from flexible polymer fibers,” ACS Photonics 1(1), 11–16 (2014).
[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), 233–239 (2013).
[Crossref] [PubMed]

Vallini, F.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

Vollmer, F.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[Crossref] [PubMed]

Wang, H.-X.

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

Wang, Q.

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

Wang, Y.

S. Yang, Y. Wang, and H. Sun, “Advances and prospects for whispering gallery mode microcavities,” Adv. Opt. Materials 3(9), 1136–1162 (2015).
[Crossref]

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

White, I. M.

Wittek, S.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

Wu, L.-H.

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

Xiao, M.

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

Xu, T.

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

Xu, Y. F.

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[Crossref] [PubMed]

Yang, S.

S. Yang, Y. Wang, and H. Sun, “Advances and prospects for whispering gallery mode microcavities,” Adv. Opt. Materials 3(9), 1136–1162 (2015).
[Crossref]

Yang, Y.

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci. Rep. 8(1), 1588 (2018).
[Crossref] [PubMed]

L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
[Crossref]

Ye, L.

L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
[Crossref]

Yves, S.

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (2017).
[Crossref] [PubMed]

Zhang, S.-C.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[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(1), 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(1), 5782 (2014).
[Crossref] [PubMed]

Zhu, S.

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

ACS Photonics (1)

R. Chen, V. D. Ta, and H. Sun, “Bending-induced bidirectional tuning of whispering gallery mode lasing from flexible polymer fibers,” ACS Photonics 1(1), 11–16 (2014).
[Crossref]

Adv. Opt. Materials (1)

S. Yang, Y. Wang, and H. Sun, “Advances and prospects for whispering gallery mode microcavities,” Adv. Opt. Materials 3(9), 1136–1162 (2015).
[Crossref]

Adv. Opt. Photonics (1)

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

L. Ye, Y. Yang, Z. Hong Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017).
[Crossref]

C.-Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

Nat. Commun. (2)

S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, “Crystalline metamaterials for topological properties at subwavelength scales,” Nat. Commun. 8, 16023 (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(1), 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), 233–239 (2013).
[Crossref] [PubMed]

Nat. Methods (1)

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008).
[Crossref] [PubMed]

Nat. Photonics (2)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[Crossref]

Nat. Phys. (1)

C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
[Crossref]

Nature (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

New J. Phys. (1)

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

Opt. Lett. (2)

Phys. Rev. B (1)

G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
[Crossref]

Phys. Rev. Lett. (7)

Y. Yang, Y. F. Xu, T. Xu, H.-X. Wang, J.-H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref] [PubMed]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[Crossref] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[Crossref] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[Crossref] [PubMed]

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Phys. Rev. X (1)

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

Sci. Rep. (1)

Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci. Rep. 8(1), 1588 (2018).
[Crossref] [PubMed]

Science (3)

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017).
[Crossref] [PubMed]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), 4005 (2018).

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314(5806), 1757–1761 (2006).
[Crossref] [PubMed]

Other (2)

K. Sakoda, Optical properties of photonic crystals (Springer Science & Business Media, 2004).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton university press, 2011).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1 (a) Schematic picture of 2D finite topological PC cavity with the lattice constant a, the diameter of dielectric cylinders d = 0.24a and relative permittivity ε = 7.5. The radius of hexagonal topological PC1 cavity (in red) is R = 3a surrounding by trivial PC2 (in blue). (b) Discrete topological interface states (red dots) of PCs labeled by numbers calculated with R = 3a (left panel), R = 2a (middle panel) and R = 6a (right panel) respectively. (c) Nearly degenerate Ez eigenfield distributions of discrete WGMs of R = 3a cavity. The red and blue lines along the circumference of the topological cavity indicate “+” and “-” eigen-field distributions respectively. Two enlarged views next to the first-order mode distribution show phase differences at the topological cavity circumference (white dashed lines).
Fig. 2
Fig. 2 (a) Eigenfield distributions of topological interface states of PCs at different shapes with identical normalized frequency and responding topological discrete states shown in (b).
Fig. 3
Fig. 3 (a) Photo of experimental setup to observe topological WGMs in PC cavity by alumina cylinders with relative permittivity ε = 7.5, height h = 8 mm, diameter d = 6 mm and lattice constant a = 25 mm. The intra (inter) couple distance is h1 = 9 mm (h2 = 7 mm) for topological PC1 (orange region), while h1 = 7.5 mm (h2 = 10 mm) for trivial PC2. A chiral source shown in the inset is used to excite WGMs. (b) Total electric field intensity calculated and discrete modes consistent to eigenmode analysis are found. (c) Measured total electric field intensity spectrum by an integration of measured electric field amplitudes of all frequencies interested. (d) Measured electric field distributions of topological WGMs at 7.531 (upper panel) and 7.709 GHz (lower panel).
Fig. 4
Fig. 4 The simulated results of an optical directional coupler. (a) At 561 THz, left-incident wave can couple to the topological cavity. The schematic diagram is shown in the left panel. PCs are composed of silicon dielectric cylinders with lattice constant a = 0.25 μm, relative permittivity ε = 12.5 and diameter d = 0.06 μm. (b) The WGM can couple to the right output if pseudospin up mode is excited at 539 THz.

Equations (1)

Equations on this page are rendered with MathJax. Learn more.

k // (ω)L=2mπ

Metrics