Abstract

The band structure of a two-dimensional photonic crystal (PhC) formed by a triangular lattice of air columns in a gyromagnetic material is investigated in the case when an external dc magnetic field is applied to it. It is shown that large magnetic-field-induced (MFI) bandgap is obtainable by optimizing the parameters of the PhC. The interface between a PhC with MFI bandgap and air may support unidirectional or bidirectional propagating edge modes, or even no mode, closely depending on the boundary shape of the truncated PhC. The transmission property of one-way mode sustained by the gyromagnetic PhC boundary is discussed through numerical simulation.

© 2013 Optical Society of America

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  1. H. Zhu and C. Jiang, “Extraordinary coupling into one-way magneto-optical photonic crystal waveguide,” J. Lightwave Technol. 29, 708–713 (2011).
    [CrossRef]
  2. X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
    [CrossRef]
  3. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).
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    [CrossRef]
  5. R. E. Prange and S. M. Girvin, The Quantum Hall Effect, ed. (Springer-Verlag, 1987).
  6. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
    [CrossRef]
  7. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
    [CrossRef]
  8. J. X. Fu, R. J. Liu, and Z. Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
    [CrossRef]
  9. Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
    [CrossRef]
  10. C. He, X. L. Chen, M. H. Lu, and X. F. Li, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96, 111111 (2010).
    [CrossRef]
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    [CrossRef]
  12. S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97, 201113 (2010).
    [CrossRef]
  13. X. F. Zang and C. Jiang, “Edge mode in nonreciprocal photonic crystal waveguide: manipulating the unidirectional electromagnetic pulse dynamically,” J. Opt. Soc. Am. B 28, 554–557 (2011).
    [CrossRef]
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    [CrossRef]
  15. D. M. Pozar, Microwave Engineering (Wiley, 1998).

2012

2011

X. F. Zang and C. Jiang, “Edge mode in nonreciprocal photonic crystal waveguide: manipulating the unidirectional electromagnetic pulse dynamically,” J. Opt. Soc. Am. B 28, 554–557 (2011).
[CrossRef]

H. Zhu and C. Jiang, “Extraordinary coupling into one-way magneto-optical photonic crystal waveguide,” J. Lightwave Technol. 29, 708–713 (2011).
[CrossRef]

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” 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, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

2010

S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97, 201113 (2010).
[CrossRef]

C. He, X. L. Chen, M. H. Lu, and X. F. Li, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

J. X. Fu, R. J. Liu, and Z. Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

2009

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

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[CrossRef]

2008

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Ao, X.

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Chan, C. T.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Chen, X. L.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

Chong, Y. D.

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Chui, S. T.

S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97, 201113 (2010).
[CrossRef]

Fu, J. X.

J. X. Fu, R. J. Liu, and Z. Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

Girvin, S. M.

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

Haldane, F. D. M.

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

He, C.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

He, S.

Jiang, C.

Joannopoulos, J. D.

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Li, X. F.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

Li, Z. Y.

J. X. Fu, R. J. Liu, and Z. Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

Lin, Z.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Lin, Z. F.

S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97, 201113 (2010).
[CrossRef]

Liu, K. X.

Liu, R. J.

J. X. Fu, R. J. Liu, and Z. Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

Liu, S. Y.

S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97, 201113 (2010).
[CrossRef]

Lu, M. H.

C. He, X. L. Chen, M. H. Lu, and X. F. Li, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

Lu, W. L.

S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97, 201113 (2010).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Poo, Y.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

Pozar, D. M.

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

Prange, R. E.

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

Raghu, S.

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

Shen, L. F.

K. X. Liu, L. F. Shen, and S. He, “One-way edge mode in a gyromagnetic photonic crystal slab,” Opt. Lett. 37, 4110–4112 (2012).
[CrossRef]

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Soljacic, M.

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Wang, Y. G.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Wang, Z.

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” 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, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Wu, R. X.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

Yang, Y.

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” 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, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Zang, X. F.

Zhang, X. M.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” 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, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

Zhu, H.

Appl. Phys. Lett.

J. X. Fu, R. J. Liu, and Z. Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[CrossRef]

C. He, X. L. Chen, M. H. Lu, and X. F. Li, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96, 111111 (2010).
[CrossRef]

S. Y. Liu, W. L. Lu, Z. F. Lin, and S. T. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97, 201113 (2010).
[CrossRef]

J. Appl. Phys.

Z. Y. Wang, L. F. Shen, X. M. Zhang, Y. G. Wang, Z. H. Yu, and X. D. Zheng, “Photonic crystal cavity with one-way rotating state and its coupling with photonic crystal waveguide,” J. Appl. Phys. 110, 043106 (2011).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Nature

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

Opt. Lett.

Phys. Rev. B

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[CrossRef]

Phys. Rev. Lett.

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

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Y. Poo, R. X. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[CrossRef]

Other

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

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

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

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Figures (6)

Fig. 1.
Fig. 1.

Band diagram (solid lines) of the YIG PhC with r=0.44a calculated with the FEM. (a) Without external magnetic field and (b) with an external magnetic field of H0=1600G in the +z direction. Circles correspond to the results for the fourth and fifth bands obtained with the PWE method. Inset in (b) shows a schematic of the PhC.

Fig. 2.
Fig. 2.

Various MFI bandgaps as a function of the radius of air columns. Gaps 1–4 correspond to the bandgaps between bands 1 and 2, between bands 3 and 4, between bands 4 and 5, and between bands 5 and 6, respectively.

Fig. 3.
Fig. 3.

(a) Truncation of the YIG PhC along the x direction at the different locations of yb=0.22a, 0, 0.22a, 0.426a, and 0.44a. (b) Dispersion relation of edge mode at different YIG PhC boundaries. Dashed–dotted, solid, dashed, and dotted lines correspond to the boundaries with yb=0.22a, 0.22a, 0.426a, and 0.44a, respectively. Solid and hollow circles represent the results of the 3D systems for yb=0.22a and 0.426a, in which the 2D system is truncated in the z direction and sandwiched by two metal slabs. The PhC slab has a thickness of h=0.5a in the 3D system. The middle shaded area represents light cone and the side ones the projected band diagram of the YIG PhC along the ΓK direction under H0=1600G.

Fig. 4.
Fig. 4.

Spatial variation of Ez field amplitude. Blue and red represent zero and large values, respectively. A line current operating at 4.28 GHz is placed at the edge of the YIG PhC. (a) PhC edge with yb=0.22a, (b) PhC edge with yb=0.426a, and (c) PhC edge with yb=0.

Fig. 5.
Fig. 5.

Spatial variation of Ez field amplitude. A line current operating at 4.28 GHz is placed at the edge of the YIG PhC. (a) PhC edge of yb=0.22a with an imperfectness. (b) A metal slab is further added at a distance of d=0.6a from the PhC edge.

Fig. 6.
Fig. 6.

Dispersion relation of one-way mode in the presence of a PEC slab terminating the air cladding. Thin solid, dashed–dotted, dashed, and dotted lines correspond to the thicknesses of air cladding of d=0.2a, 0.4a, 0.6a, and 1a, respectively. Thick solid line represents the dispersion relation for the case without PEC slab.

Equations (3)

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μ¯=(μriμk0iμkμr000μ0),
μr=1+ωmωoω02ω2,
μk=ωmωω02ω2,

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