Abstract

We demonstrate that one-way electromagnetic modes could be sustained by the edge of a gyromagnetic photonic crystal slab of triangular lattice under an external dc magnetic field. The applied magnetic field breaks the time-reversal symmetry of the three-dimensional system, and thus the original degeneracy point in k space, at which two dispersion surfaces intersect, is lifted, resulting in a photonic band gap below the light cone. At this band gap, the one-way mode is localized horizontally to the slab edge, while confined by the index contrast in the vertical direction.

© 2012 Optical Society of America

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef]
  2. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef]
  3. A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
    [CrossRef]
  4. F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
    [CrossRef]
  5. R. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer-Verlag, 1987).
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    [CrossRef]
  7. X. Ao, Z. Lin, and C. T. Chan, Phys. Rev. B 80, 033105 (2009).
    [CrossRef]
  8. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).
    [CrossRef]
  9. J. X. Fu, R. J. Liu, and Z. Y. Li, Appl. Phys. Lett. 97, 041112 (2010).
    [CrossRef]
  10. Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
    [CrossRef]
  11. C. He, X. L. Chen, M. H. Lu, and X. F. Li, Appl. Phys. Lett. 96, 111111 (2010).
    [CrossRef]
  12. Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
    [CrossRef]
  13. H. Takeda and S. John, Phys. Rev. A 78, 023804 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. H. B. Zhu and C. Jiang, Opt. Express 18, 6914 (2010).
    [CrossRef]
  17. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  18. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
    [CrossRef]
  19. D. M. Pozar, Microwave Engineering (Wiley, 1998).
  20. T. Weiland, Electron. Commun. 31, 116 (1977).

2011 (2)

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

2010 (4)

C. He, X. L. Chen, M. H. Lu, and X. F. Li, Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, Phys. Rev. Lett. 105, 126804 (2010).
[CrossRef]

H. B. Zhu and C. Jiang, Opt. Express 18, 6914 (2010).
[CrossRef]

J. X. Fu, R. J. Liu, and Z. Y. Li, Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

2009 (3)

X. Ao, Z. Lin, and C. T. Chan, Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).
[CrossRef]

A. B. Khanikaev, A. V. Baryshev, M. Inoue, and Y. S. Kivshar, Appl. Phys. Lett. 95, 011101 (2009).
[CrossRef]

2008 (3)

H. Takeda and S. John, Phys. Rev. A 78, 023804 (2008).
[CrossRef]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

2003 (1)

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

1999 (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
[CrossRef]

1987 (2)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

1977 (1)

T. Weiland, Electron. Commun. 31, 116 (1977).

Ao, X.

X. Ao, Z. Lin, and C. T. Chan, Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Baryshev, A. V.

A. B. Khanikaev, A. V. Baryshev, M. Inoue, and Y. S. Kivshar, Appl. Phys. Lett. 95, 011101 (2009).
[CrossRef]

Chan, C. T.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

X. Ao, Z. Lin, and C. T. Chan, Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Chen, X. L.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Fu, J. X.

J. X. Fu, R. J. Liu, and Z. Y. Li, Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

He, C.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

Inoue, M.

A. B. Khanikaev, A. V. Baryshev, M. Inoue, and Y. S. Kivshar, Appl. Phys. Lett. 95, 011101 (2009).
[CrossRef]

Jiang, C.

Joannopoulos, J. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

John, S.

H. Takeda and S. John, Phys. Rev. A 78, 023804 (2008).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Johnson, S. G.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
[CrossRef]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, Phys. Rev. Lett. 105, 126804 (2010).
[CrossRef]

A. B. Khanikaev, A. V. Baryshev, M. Inoue, and Y. S. Kivshar, Appl. Phys. Lett. 95, 011101 (2009).
[CrossRef]

Kivshar, Y. S.

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, Phys. Rev. Lett. 105, 126804 (2010).
[CrossRef]

A. B. Khanikaev, A. V. Baryshev, M. Inoue, and Y. S. Kivshar, Appl. Phys. Lett. 95, 011101 (2009).
[CrossRef]

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
[CrossRef]

Li, X. F.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

Li, Z. Y.

J. X. Fu, R. J. Liu, and Z. Y. Li, Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

Lin, Z.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

X. Ao, Z. Lin, and C. T. Chan, Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Liu, R. J.

J. X. Fu, R. J. Liu, and Z. Y. Li, Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

Lu, M. H.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, Phys. Rev. Lett. 105, 126804 (2010).
[CrossRef]

Poo, Y.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

Raghu, S.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

Shen, L. F.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, Phys. Rev. Lett. 105, 126804 (2010).
[CrossRef]

Soljacic, M.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Takeda, H.

H. Takeda and S. John, Phys. Rev. A 78, 023804 (2008).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
[CrossRef]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Wang, Y. G.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Wang, Z.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Wang, Z. Y.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Weiland, T.

T. Weiland, Electron. Commun. 31, 116 (1977).

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Wu, R. X.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

Yang, Y.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

Yu, Z. H.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Zhang, X. M.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Zheng, X. D.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Zhu, H. B.

Appl. Phys. Lett. (3)

J. X. Fu, R. J. Liu, and Z. Y. Li, Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

C. He, X. L. Chen, M. H. Lu, and X. F. Li, Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

A. B. Khanikaev, A. V. Baryshev, M. Inoue, and Y. S. Kivshar, Appl. Phys. Lett. 95, 011101 (2009).
[CrossRef]

Electron. Commun. (1)

T. Weiland, Electron. Commun. 31, 116 (1977).

J. Appl. Phys. (1)

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Nature (1)

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

H. Takeda and S. John, Phys. Rev. A 78, 023804 (2008).
[CrossRef]

Phys. Rev. B (3)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
[CrossRef]

X. Ao, Z. Lin, and C. T. Chan, Phys. Rev. B 80, 033105 (2009).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Phys. Rev. Lett. (6)

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, Phys. Rev. Lett. 105, 126804 (2010).
[CrossRef]

Other (3)

D. M. Pozar, Microwave Engineering (Wiley, 1998).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

R. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer-Verlag, 1987).

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Figures (4)

Fig. 1.
Fig. 1.

Gyromagnetic PhC slab formed by a triangular lattice of holes in a YIG slab surrounded by air. With discrete translational symmetry in the xy plane, the in-plane wave vector is conserved. The right panel shows the first Brillouin zone.

Fig. 2.
Fig. 2.

Band structure for a triangular lattice of air holes in a YIG slab with εr=15, r=0.45a, h=0.8a and a=11mm. The gray shaded area is the light cone, all of extended modes propagating in air. (a) Without dc magnetic field applied. (b) With 1700 Gauss magnetic field applied in the +z direction.

Fig. 3.
Fig. 3.

Projected band structure (blue dots) along the ΓK direction for a triangular lattice of air holes in a YIG slab under 1700 Gauss magnetic field applied in +z direction. The solid (red) line in the band gap is the dispersion curve of the edge mode. The gray shaded area is the light cone.

Fig. 4.
Fig. 4.

Spatial variation of Ez field amplitude. Blue and red represent zero and large values respectively. A line current operating at 10 GHz is placed at the edge of the YIG PhC slab. (a) Horizontal cross-section at the mid-height of the YIG slab. (b) Vertical cross-section at y=r. (c), (d) Horizontal cross-sections at the mid-height of the PhC slab with imperfect periodic edge, and in (d) a metal slab of thickness H=4h is further added by the PhC slab edge. (e) Ez amplitude along the x axis for the cases of (a) (solid), (c) (dotted), and (d) (dashed).

Equations (1)

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μ¯=[μriμk0iμkμr0001],

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