Abstract

A general coupled-wave theory is presented for the guided resonance in photonic crystal (PhC) slabs with TM-like polarization. Numerical results based on our model are presented with finite-difference time-domain validations. The proposed analysis facilitates comprehensive understanding of the physics of guided resonance in PhC slabs and provides guidance for its applications.

© 2014 Optical Society of America

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  1. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
    [CrossRef]
  2. K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
    [CrossRef]
  3. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Phys. Rev. B 60, 5751 (1999).
    [CrossRef]
  4. J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
    [CrossRef]
  5. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
    [CrossRef]
  6. A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
    [CrossRef]
  7. W. Streifer, D. Scifres, and R. Burnham, IEEE J. Quantum Electron. 12, 74 (1976).
    [CrossRef]
  8. Y. Yamamoto, T. Kamiya, and H. Yanai, IEEE J. Quantum Electron. 14, 620 (1978).
    [CrossRef]
  9. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 84, 195119 (2011).
    [CrossRef]
  10. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 86, 035108 (2012).
    [CrossRef]
  11. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 19, 24672 (2011).
    [CrossRef]
  12. Y. Yang, C. Peng, and Z. Li, Opt. Express 21, 20588 (2013).
    [CrossRef]
  13. Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, Opt. Express 21, 565 (2013).
    [CrossRef]
  14. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 20, 15945 (2012).
    [CrossRef]
  15. S. Chua, L. Lu, J. Bravo-Abad, J. Joannopoulos, and M. Soljačić, Opt. Lett. 39, 2072 (2014).
    [CrossRef]
  16. Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
    [CrossRef]
  17. M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
    [CrossRef]
  18. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  19. Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys. Rev. Lett. 113, 037401 (2014).

2014 (3)

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

S. Chua, L. Lu, J. Bravo-Abad, J. Joannopoulos, and M. Soljačić, Opt. Lett. 39, 2072 (2014).
[CrossRef]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys. Rev. Lett. 113, 037401 (2014).

2013 (3)

2012 (4)

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 20, 15945 (2012).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 86, 035108 (2012).
[CrossRef]

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

2011 (2)

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 19, 24672 (2011).
[CrossRef]

2009 (1)

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

2006 (1)

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[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]

1978 (1)

Y. Yamamoto, T. Kamiya, and H. Yanai, IEEE J. Quantum Electron. 14, 620 (1978).
[CrossRef]

1976 (1)

W. Streifer, D. Scifres, and R. Burnham, IEEE J. Quantum Electron. 12, 74 (1976).
[CrossRef]

1973 (1)

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

Asano, T.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

Barbieri, S.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Beere, H. E.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Bravo-Abad, J.

Burnham, R.

W. Streifer, D. Scifres, and R. Burnham, IEEE J. Quantum Electron. 12, 74 (1976).
[CrossRef]

Chassagneux, Y.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Chua, S.

Chua, S. L.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[CrossRef]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Colombelli, R.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Davies, A. G.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[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]

Hirose, K.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

Hsu, C. W.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[CrossRef]

Inoue, T.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

Ishizaki, K.

Iwahashi, S.

Joannopoulos, J.

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[CrossRef]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[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, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S. G.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[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, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kamiya, T.

Y. Yamamoto, T. Kamiya, and H. Yanai, IEEE J. Quantum Electron. 14, 620 (1978).
[CrossRef]

Khanna, S. P.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Kitamura, K.

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]

Kunishi, W.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Kurosaka, Y.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[CrossRef]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Li, Z.

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys. Rev. Lett. 113, 037401 (2014).

Y. Yang, C. Peng, and Z. Li, Opt. Express 21, 20588 (2013).
[CrossRef]

Liang, Y.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys. Rev. Lett. 113, 037401 (2014).

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, Opt. Express 21, 565 (2013).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 20, 15945 (2012).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 19, 24672 (2011).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 84, 195119 (2011).
[CrossRef]

Linfield, E. H.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Lu, L.

Maineult, W.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Meade, R. D.

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

Miyai, E.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Mochizuki, K.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

Noda, A.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

Noda, S.

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys. Rev. Lett. 113, 037401 (2014).

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, Opt. Express 21, 565 (2013).
[CrossRef]

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 20, 15945 (2012).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 19, 24672 (2011).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 84, 195119 (2011).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Ohnishi, D.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Okano, T.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Oskooi, A.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

Peng, C.

Qiu, W.

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Ritchie, D. A.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

Sakai, K.

Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, Opt. Express 21, 565 (2013).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 20, 15945 (2012).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Opt. Express 19, 24672 (2011).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 84, 195119 (2011).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Scifres, D.

W. Streifer, D. Scifres, and R. Burnham, IEEE J. Quantum Electron. 12, 74 (1976).
[CrossRef]

Shapira, O.

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Soljacic, M.

S. Chua, L. Lu, J. Bravo-Abad, J. Joannopoulos, and M. Soljačić, Opt. Lett. 39, 2072 (2014).
[CrossRef]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[CrossRef]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Streifer, W.

W. Streifer, D. Scifres, and R. Burnham, IEEE J. Quantum Electron. 12, 74 (1976).
[CrossRef]

Sugiyama, T.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

Tanaka, Y.

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]

Watanabe, A.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

Winn, J. N.

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

Yamamoto, Y.

Y. Yamamoto, T. Kamiya, and H. Yanai, IEEE J. Quantum Electron. 14, 620 (1978).
[CrossRef]

Yanai, H.

Y. Yamamoto, T. Kamiya, and H. Yanai, IEEE J. Quantum Electron. 14, 620 (1978).
[CrossRef]

Yang, Y.

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys. Rev. Lett. 113, 037401 (2014).

Y. Yang, C. Peng, and Z. Li, Opt. Express 21, 20588 (2013).
[CrossRef]

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

Zhen, B.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[CrossRef]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Zoysa, M. D.

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

IEEE J. Quantum Electron. (3)

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

W. Streifer, D. Scifres, and R. Burnham, IEEE J. Quantum Electron. 12, 74 (1976).
[CrossRef]

Y. Yamamoto, T. Kamiya, and H. Yanai, IEEE J. Quantum Electron. 14, 620 (1978).
[CrossRef]

Nat. Photonics (2)

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and A. Noda, Nat. Photonics 8, 406 (2014).
[CrossRef]

M. D. Zoysa, T. Asano, K. Mochizuki, A. Oskooi, T. Inoue, and S. Noda, Nat. Photonics 6, 535 (2012).
[CrossRef]

Nature (3)

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, Nature 457, 174 (2009).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Nature 499, 188 (2013).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

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]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Phys. Rev. Lett. (2)

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, Phys. Rev. Lett. 109, 067401 (2012).
[CrossRef]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys. Rev. Lett. 113, 037401 (2014).

Other (1)

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

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Figures (4)

Fig. 1.
Fig. 1.

(a) A photonic crystal slab in real space. (b) Wavevectors in k space. Basic waves (blue, Rx, Sx, Ry, Sy) and high-order waves (orange) are considered [914]. (c) Typical TM-like band structure near the second-order Γ point with four identified band-edge modes [9].

Fig. 2.
Fig. 2.

CWT-calculated in-plane H-field vector distribution (Hx, Hy arrows) and Ez patterns (in color) of four band-edge modes (TM1TM4) at the 3D structure. The thick black circles indicate the locations of holes.

Fig. 3.
Fig. 3.

Example (structure in [5]): modal frequency and loss of TM1TM4 modes versus the filling factor.

Fig. 4.
Fig. 4.

Example (structure in [5]): modal properties in the vicinity of the second-order Γ point.

Equations (16)

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

×[1ϵ(r)×H(r)]=k02H(r).
(κa2z2+κaδ00z+k02κany2β02)Hx,m,n+κamxnyβ02Hy,m,n=m,nm,nκmmnn,[(2z2+δmmnn,z+nynyβ02)Hx,m,nmxnyβ02Hy,m,n],
m,nκmmnn,z(mxHx,m,n+nyHy,m,n)=0.
[2z2δ00z+(mx2+ny2)β02]κaΘm,n=k0;mn2Θm,n.
(k0;mn2k02)mx2+ny2AmnΘm,n=mmnn,κmmnn,[2z2δmmnn,z(mx2+ny2)β02](nyHx,m,nmxHy,m,n),
J^1ΔHx=m,n0κmn,K^mnHx,m,n,
ΔHx=m,n0κmn,G1(z,z)K^Hx,mn(z)dz,
J^2(mxHx,m,n+nyHy,m,n)J^2H+=mmnn,κmmnn,K^(mxHx,m,n+nyHy,m,n),
U^mn(nyHx,m,nmxHy,m,n)U^mnH=mmnn,κmmnn,V^mn(nyHx,m,nmxHy,m,n),
C=k0,mn+C1D+Crad+C2D,
k0,mn=(k0;1,00000k0;1,00000k0;0,10000k0;0,1),
C1D=(0κ2,000κ2,0000000κ0,200κ0,20),
Crad=(ζ1,0(1,0)ζ1,0(1,0)ζ1,0(0,1)ζ1,0(0,1)ζ1,0(1,0)ζ1,0(1,0)ζ1,0(0,1)ζ1,0(0,1)ζ0,1(1,0)ζ0,1(1,0)ζ0,1(0,1)ζ0,1(0,1)ζ0,1(1,0)ζ0,1(1,0)ζ0,1(0,1)ζ0,1(0,1)),
C2D=(χ1,0(1,0)χ1,0(1,0)χ1,0(0,1)χ1,0(0,1)χ1,0(1,0)χ1,0(1,0)χ1,0(0,1)χ1,0(0,1)χ0,1(1,0)χ0,1(1,0)χ0,1(0,1)χ0,1(0,1)χ0,1(1,0)χ0,1(1,0)χ0,1(0,1)χ0,1(0,1)).
ζmn(rs)=k042βmnκm,nκr,sV^0,0[Gx,0,0(z,z)Gy,0,0(z,z)]Θm,n(z)Θr,s*(z)dzdz,
χmn(rs)=k042βmn|m2+n2|>1κmmnn,κmrns,V^m,n(nyGx,m,nmxGy,m,n)Θm,n(z)Θr,s*(z)dzdz,

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