Abstract

We analyze the origin of the fake modes introduced by the plane wave expansion method with three-dimension (3D) supercell approximation. Through the detailed analysis of the energy distribution of fake modes and real modes, we propose the plane wave expansion–three planar-slab waveguides method to remove the fake modes and obtain the fake mode free band structure of a two-dimensional air hole photonic crystal slab. To the best of our knowledge, this is the first time that such a fake mode free photonic crystal band structure is presented. Our method is also definitely useful in designing other 3D devices.

© 2011 Optical Society of America

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  1. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2010 (1)

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

2003 (1)

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

2002 (2)

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Z. M. Zhu and T. G. Brown, Opt. Express 10, 853(2002).
[PubMed]

2001 (3)

2000 (2)

S. Noda, A. Chutinan, and M. Imada, Nature 407, 608(2000).
[CrossRef] [PubMed]

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

1999 (2)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

M. Qiu and S. L. He, Phys. Rev. B 60, 10610 (1999).
[CrossRef]

1993 (1)

1991 (1)

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Azizi, K.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Brommer, K. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Brown, T. G.

Chan, C. T.

W. Zhang, C. T. Chan, and P. Sheng, Opt. Express 8, 203 (2001).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Chen, W.

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

Chutinan, A.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, Phys. Rev. B 65, 195306 (2002).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, Nature 407, 608(2000).
[CrossRef] [PubMed]

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Dalichaouch, R.

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

He, S. L.

M. Qiu and S. L. He, Phys. Rev. B 60, 10610 (1999).
[CrossRef]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Imada, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, Phys. Rev. B 65, 195306 (2002).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, Nature 407, 608(2000).
[CrossRef] [PubMed]

Jaskorzynska, B.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Joannopoulos, J. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Karlsson, A.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Kroll, N.

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Li, Z. Y.

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, L. L.

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Liu, A. J.

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

McCall, S. L.

Meade, R. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Mochizuki, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Modinos, A.

Noda, S.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, Phys. Rev. B 65, 195306 (2002).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, Nature 407, 608(2000).
[CrossRef] [PubMed]

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Platzman, P. M.

Qiu, M.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, Phys. Rev. B 64, 155113 (2001).
[CrossRef]

M. Qiu and S. L. He, Phys. Rev. B 60, 10610 (1999).
[CrossRef]

Qu, H. W.

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

Rappe, A. M.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals(Springer Verlag, 2001).

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Schultz, S.

Sheng, P.

Smith, D. R.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Stefanou, N.

Swillo, M.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, Phys. Rev. B 64, 155113 (2001).
[CrossRef]

Xing, M. X.

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Yannopapas, V.

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Zhang, W.

Zheng, W. H.

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

Zhou, W. J.

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

Zhu, Z. M.

Appl. Phys. Lett. (1)

A. J. Liu, W. Chen, M. X. Xing, W. J. Zhou, H. W. Qu, and W. H. Zheng, Appl. Phys. Lett. 96, 151103 (2010).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

S. Noda, A. Chutinan, and M. Imada, Nature 407, 608(2000).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Rev. B (4)

M. Qiu and S. L. He, Phys. Rev. B 60, 10610 (1999).
[CrossRef]

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, Phys. Rev. B 64, 155113 (2001).
[CrossRef]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Phys. Rev. E (1)

Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Science (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 (1999).
[CrossRef] [PubMed]

Other (1)

K. Sakoda, Optical Properties of Photonic Crystals(Springer Verlag, 2001).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic structure of 2D air hole PhC slab. (b) Supercell used in the PWE method with 3D SCA.

Fig. 2
Fig. 2

(a) 3D magnetic field of a fake mode Γ fake ( f = 0.29687 ( 2 π c / a ) ) calculated by 3D PWE method with SCA. (b) Magnetic field of a mode with frequency around f fake calculated by FDTD method. (c) Band structure of 2D air hole PhC slab calculated by 3D PWE method with SCA. (d) 3D magnetic field of a real mode Γ real ( f = 0.38829 ( 2 π c / a ) ) calculated by 3D PWE method with SCA. (e) Magnetic field of the real mode Γ real calculated by FDTD method.

Fig. 3
Fig. 3

Band structures calculated by PWE method and SCA with different supercells. (a)  1 a × 1 a × 9 a ,(b)  1 a × 1 a × 21 a .

Fig. 4
Fig. 4

(a) TPSW model. (b) and (c) are the normalized intensities along the z direction calculated by the TPSW model. The red lines indicate the upper and lower boundaries of the slab.

Fig. 5
Fig. 5

(a) and (b)  W slab , TPSW for TE and TM modes calculated by the TPSW model. (c) Part of even mode band structure for PhC slab. (d) Part of Fig. 5c along Γ X direction with marked ( W slab , PWE , W slab , TPSW ). (e) Fake mode free band structure of 2D air hole PhC slab.

Equations (1)

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n eff = 2.1172 ,

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