Abstract

We numerically calculated the photonic band structure of a cholesteric blue phase II (BPII) liquid crystal using two models, the tensor order parameter model and the double-twist cylinder (DTC) model. The tensor order parameter model was derived numerically from the Landau–de Gennes theory, whereas the DTC model simply assumed a three-dimensional stack of ordered cylinders. The two models yielded similar photonic band structures; however, the polarization dependence of the transmission spectrum through the (100) direction in BPII was different. The results from the tensor order parameter model gave a better qualitative agreement with experiments on the transmission spectrum. This demonstrates the importance of a precise orientation model in the analysis of the optical properties of cholesteric blue phases.

© 2013 Optical Society of America

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    [CrossRef]
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  8. Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S.-T. Wu, Appl. Phys. Lett. 94, 101104 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. J. P. Berenger, J. Comput. Phys. 114, 185 (1994).
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  21. J. H. Flack and P. P. Crooker, Phys. Lett. A 82, 247 (1981).
    [CrossRef]
  22. J. Fukuda, Phys. Rev. E 82, 061702 (2010).
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  23. J. Fukuda, Phys. Rev. E 86, 041704 (2012).
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    [CrossRef]

2012 (2)

J. Fukuda, Phys. Rev. E 86, 041704 (2012).
[CrossRef]

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

2011 (1)

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Proc. Natl. Acad. Sci. 108, 5188 (2011).
[CrossRef]

2010 (4)

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Faraday Discuss. 144, 159 (2010).
[CrossRef]

Y.-H. Lin, H.-S. Chen, H.-C. Lin, Y.-S. Tsou, H.-K. Hsu, and W.-Y. Li, Appl. Phys. Lett. 96, 113505 (2010).
[CrossRef]

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

J. Fukuda, Phys. Rev. E 82, 061702 (2010).
[CrossRef]

2009 (3)

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

J. Fukuda, M. Yoneya, and H. Yokoyama, Phys. Rev. E 80, 031706 (2009).
[CrossRef]

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S.-T. Wu, Appl. Phys. Lett. 94, 101104 (2009).
[CrossRef]

2005 (1)

Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, Adv. Mater. 17, 96 (2005).
[CrossRef]

2004 (1)

C. Bohley and T. Scharf, J. Opt. A 6, S77 (2004).

2003 (1)

S. Urban, B. Gestblom, W. Kuczynski, S. Pawlus, and A. Würflinger, Phys. Chem. Chem. Phys. 5, 924 (2003).
[CrossRef]

2002 (2)

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, Nat. Mater. 1, 64 (2002).
[CrossRef]

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, Nat. Mater. 1, 111 (2002).
[CrossRef]

1994 (1)

J. P. Berenger, J. Comput. Phys. 114, 185 (1994).
[CrossRef]

1993 (1)

R. M. Hornreich, S. Shtrikman, and C. Sommers, Phys. Rev. E 47, 2067 (1993).
[CrossRef]

1989 (1)

D. C. Wright and N. D. Mermin, Rev. Mod. Phys. 61, 385 (1989).
[CrossRef]

1983 (2)

H. Grebel, R. M. Hornreich, and S. Shtrikman, Phys. Rev. A 28, 1114 (1983).
[CrossRef]

S. Meiboom, M. Sammon, and W. F. Brinkman, Phys. Rev. A 27, 438 (1983).
[CrossRef]

1981 (1)

J. H. Flack and P. P. Crooker, Phys. Lett. A 82, 247 (1981).
[CrossRef]

1980 (1)

D. L. Johnson, J. H. Flack, and P. P. Crooker, Phys. Rev. Lett. 45, 641 (1980).
[CrossRef]

Alexander, G. P.

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Proc. Natl. Acad. Sci. 108, 5188 (2011).
[CrossRef]

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Faraday Discuss. 144, 159 (2010).
[CrossRef]

Belyakov, V. A.

V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, 1992), Chap. 4.

V. A. Belyakov and V. E. Dmitrienko, Optics of Chiral Liquid Crystals, I. M. Khalatnikov, ed., Vol. 13 of Soviet Scientific Reviews/Section A (Harwood Academic, 1989).

Berenger, J. P.

J. P. Berenger, J. Comput. Phys. 114, 185 (1994).
[CrossRef]

Bohley, C.

C. Bohley and T. Scharf, J. Opt. A 6, S77 (2004).

Brinkman, W. F.

S. Meiboom, M. Sammon, and W. F. Brinkman, Phys. Rev. A 27, 438 (1983).
[CrossRef]

Cao, W.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, Nat. Mater. 1, 111 (2002).
[CrossRef]

Castles, F.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Chen, H.-S.

Y.-H. Lin, H.-S. Chen, H.-C. Lin, Y.-S. Tsou, H.-K. Hsu, and W.-Y. Li, Appl. Phys. Lett. 96, 113505 (2010).
[CrossRef]

Choi, S. S.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Coles, H. J.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Crooker, P. P.

J. H. Flack and P. P. Crooker, Phys. Lett. A 82, 247 (1981).
[CrossRef]

D. L. Johnson, J. H. Flack, and P. P. Crooker, Phys. Rev. Lett. 45, 641 (1980).
[CrossRef]

Day, F. V.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

de Gennes, P. G.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford University, 1993).

Dmitrienko, V. E.

V. A. Belyakov and V. E. Dmitrienko, Optics of Chiral Liquid Crystals, I. M. Khalatnikov, ed., Vol. 13 of Soviet Scientific Reviews/Section A (Harwood Academic, 1989).

Flack, J. H.

J. H. Flack and P. P. Crooker, Phys. Lett. A 82, 247 (1981).
[CrossRef]

D. L. Johnson, J. H. Flack, and P. P. Crooker, Phys. Rev. Lett. 45, 641 (1980).
[CrossRef]

Friend, R. H.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Fujii, A.

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Fukuda, J.

J. Fukuda, Phys. Rev. E 86, 041704 (2012).
[CrossRef]

J. Fukuda, Phys. Rev. E 82, 061702 (2010).
[CrossRef]

J. Fukuda, M. Yoneya, and H. Yokoyama, Phys. Rev. E 80, 031706 (2009).
[CrossRef]

Gardiner, D. J.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Gauza, S.

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S.-T. Wu, Appl. Phys. Lett. 94, 101104 (2009).
[CrossRef]

Ge, Z.

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S.-T. Wu, Appl. Phys. Lett. 94, 101104 (2009).
[CrossRef]

Gestblom, B.

S. Urban, B. Gestblom, W. Kuczynski, S. Pawlus, and A. Würflinger, Phys. Chem. Chem. Phys. 5, 924 (2003).
[CrossRef]

Grebel, H.

H. Grebel, R. M. Hornreich, and S. Shtrikman, Phys. Rev. A 28, 1114 (1983).
[CrossRef]

Hands, P. J. W.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Hisakado, Y.

Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, Adv. Mater. 17, 96 (2005).
[CrossRef]

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, Nat. Mater. 1, 64 (2002).
[CrossRef]

Hornreich, R. M.

R. M. Hornreich, S. Shtrikman, and C. Sommers, Phys. Rev. E 47, 2067 (1993).
[CrossRef]

H. Grebel, R. M. Hornreich, and S. Shtrikman, Phys. Rev. A 28, 1114 (1983).
[CrossRef]

Hsu, H.-K.

Y.-H. Lin, H.-S. Chen, H.-C. Lin, Y.-S. Tsou, H.-K. Hsu, and W.-Y. Li, Appl. Phys. Lett. 96, 113505 (2010).
[CrossRef]

Jiao, M.

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S.-T. Wu, Appl. Phys. Lett. 94, 101104 (2009).
[CrossRef]

Johnson, D. L.

D. L. Johnson, J. H. Flack, and P. P. Crooker, Phys. Rev. Lett. 45, 641 (1980).
[CrossRef]

Kajiyama, T.

Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, Adv. Mater. 17, 96 (2005).
[CrossRef]

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, Nat. Mater. 1, 64 (2002).
[CrossRef]

Kawamoto, K.

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Kikuchi, H.

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, Adv. Mater. 17, 96 (2005).
[CrossRef]

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, Nat. Mater. 1, 64 (2002).
[CrossRef]

Ko, D.-H.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Kubo, H.

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Kuczynski, W.

S. Urban, B. Gestblom, W. Kuczynski, S. Pawlus, and A. Würflinger, Phys. Chem. Chem. Phys. 5, 924 (2003).
[CrossRef]

Kuwabata, S.

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Li, W.-Y.

Y.-H. Lin, H.-S. Chen, H.-C. Lin, Y.-S. Tsou, H.-K. Hsu, and W.-Y. Li, Appl. Phys. Lett. 96, 113505 (2010).
[CrossRef]

Lin, H.-C.

Y.-H. Lin, H.-S. Chen, H.-C. Lin, Y.-S. Tsou, H.-K. Hsu, and W.-Y. Li, Appl. Phys. Lett. 96, 113505 (2010).
[CrossRef]

Lin, Y.-H.

Y.-H. Lin, H.-S. Chen, H.-C. Lin, Y.-S. Tsou, H.-K. Hsu, and W.-Y. Li, Appl. Phys. Lett. 96, 113505 (2010).
[CrossRef]

Meiboom, S.

S. Meiboom, M. Sammon, and W. F. Brinkman, Phys. Rev. A 27, 438 (1983).
[CrossRef]

Mermin, N. D.

D. C. Wright and N. D. Mermin, Rev. Mod. Phys. 61, 385 (1989).
[CrossRef]

Moritake, H.

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

Morris, S. M.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Muñoz, A.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, Nat. Mater. 1, 111 (2002).
[CrossRef]

Nagamura, T.

Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, Adv. Mater. 17, 96 (2005).
[CrossRef]

Nosheen, S.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Ogawa, Y.

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

Ojima, M.

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

Ozaki, M.

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Ozaki, R.

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

Palffy-Muhoray, P.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, Nat. Mater. 1, 111 (2002).
[CrossRef]

Pawlus, S.

S. Urban, B. Gestblom, W. Kuczynski, S. Pawlus, and A. Würflinger, Phys. Chem. Chem. Phys. 5, 924 (2003).
[CrossRef]

Prost, J.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford University, 1993).

Qasim, M. M.

F. Castles, F. V. Day, S. M. Morris, D.-H. Ko, D. J. Gardiner, M. M. Qasim, S. Nosheen, P. J. W. Hands, S. S. Choi, R. H. Friend, and H. J. Coles, Nat. Mater. 11, 599 (2012).
[CrossRef]

Ravnik, M.

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Proc. Natl. Acad. Sci. 108, 5188 (2011).
[CrossRef]

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Faraday Discuss. 144, 159 (2010).
[CrossRef]

Sammon, M.

S. Meiboom, M. Sammon, and W. F. Brinkman, Phys. Rev. A 27, 438 (1983).
[CrossRef]

Scharf, T.

C. Bohley and T. Scharf, J. Opt. A 6, S77 (2004).

Shtrikman, S.

R. M. Hornreich, S. Shtrikman, and C. Sommers, Phys. Rev. E 47, 2067 (1993).
[CrossRef]

H. Grebel, R. M. Hornreich, and S. Shtrikman, Phys. Rev. A 28, 1114 (1983).
[CrossRef]

Sommers, C.

R. M. Hornreich, S. Shtrikman, and C. Sommers, Phys. Rev. E 47, 2067 (1993).
[CrossRef]

Taheri, B.

W. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, Nat. Mater. 1, 111 (2002).
[CrossRef]

Tanaka, Y.

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Tsou, Y.-S.

Y.-H. Lin, H.-S. Chen, H.-C. Lin, Y.-S. Tsou, H.-K. Hsu, and W.-Y. Li, Appl. Phys. Lett. 96, 113505 (2010).
[CrossRef]

Tsuda, T.

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Urban, S.

S. Urban, B. Gestblom, W. Kuczynski, S. Pawlus, and A. Würflinger, Phys. Chem. Chem. Phys. 5, 924 (2003).
[CrossRef]

Wright, D. C.

D. C. Wright and N. D. Mermin, Rev. Mod. Phys. 61, 385 (1989).
[CrossRef]

Wu, S.-T.

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S.-T. Wu, Appl. Phys. Lett. 94, 101104 (2009).
[CrossRef]

Würflinger, A.

S. Urban, B. Gestblom, W. Kuczynski, S. Pawlus, and A. Würflinger, Phys. Chem. Chem. Phys. 5, 924 (2003).
[CrossRef]

Xianyu, H.

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S.-T. Wu, Appl. Phys. Lett. 94, 101104 (2009).
[CrossRef]

Yang, H.

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, Nat. Mater. 1, 64 (2002).
[CrossRef]

Yeomans, J. M.

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Proc. Natl. Acad. Sci. 108, 5188 (2011).
[CrossRef]

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Faraday Discuss. 144, 159 (2010).
[CrossRef]

Yokota, M.

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, Nat. Mater. 1, 64 (2002).
[CrossRef]

Yokoyama, H.

J. Fukuda, M. Yoneya, and H. Yokoyama, Phys. Rev. E 80, 031706 (2009).
[CrossRef]

Yoneya, M.

J. Fukuda, M. Yoneya, and H. Yokoyama, Phys. Rev. E 80, 031706 (2009).
[CrossRef]

Yoshida, H.

M. Ojima, Y. Ogawa, R. Ozaki, H. Moritake, H. Yoshida, A. Fujii, and M. Ozaki, Appl. Phys. Express 3, 032001 (2010).
[CrossRef]

H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, Appl. Phys. Express 2, 121501 (2009).
[CrossRef]

Zumer, S.

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Proc. Natl. Acad. Sci. 108, 5188 (2011).
[CrossRef]

M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Faraday Discuss. 144, 159 (2010).
[CrossRef]

Adv. Mater. (1)

Y. Hisakado, H. Kikuchi, T. Nagamura, and T. Kajiyama, Adv. Mater. 17, 96 (2005).
[CrossRef]

Appl. Phys. Express (2)

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the DTC model.

Fig. 2.
Fig. 2.

(a) BPII (001) plane, and dielectric constant distribution of the BPII (001) plane for (b) L-dG model and (c) DTC model.

Fig. 3.
Fig. 3.

Geometry used for analysis of the transmission spectrum along the (100) direction in BPII.

Fig. 4.
Fig. 4.

Photonic band structures of BPII for (a) L-dG model and (b) DTC model, and the dispersion relation for circular polarization in the ΓX direction near X for (c) L-dG model and (d) DTC model.

Fig. 5.
Fig. 5.

Transmission spectra of BPII in the (100) direction for (a) L-dG model and (b) DTC model.

Tables (1)

Tables Icon

Table 1. Highest and Lowest Eigenmode Frequencies in the First Group of Eigenmodes at the Edge of the Brillouin Zone

Equations (4)

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εαβ=23Δε×Qαβ+εaveδαβ,
S2=32Tr(Q2).
εe=εave+23Δε×S,
εo=εave13Δε×S.

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