Abstract

This paper presents an analytical treatment of equal-frequency surface analysis of a two-dimensional photonic crystal. We first define the equal-frequency surface in terms of plane waves, which can be numerically evaluated. Then one- and two-plane-wave approximations are proposed, which consequently lead to analytical expressions of the equal-frequency surface. The approach presented is well suited to two-dimensional photonic crystals of weak dielectric modulation. For photonic crystals with a large modulation, the approach can be used to gain a general idea of the shape of the bands.

© 2008 Optical Society of America

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  1. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  2. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  3. J. D. Joannopoulus, R. D. Meade, and J. N. Winn, Photonic Crystals Molding The Flow of Light (Princeton, 1995), pp. 94-100.
  4. J. C. Knight, T. A. Birks, P., St. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
    [CrossRef] [PubMed]
  5. T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys. Rev. B 61, 15688-15696 (2000).
    [CrossRef]
  6. B. D'Urso, O. Painter, J. O'Brien, T. Tombrello, A. Yariv, and A. Scherer, "Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities," J. Opt. Soc. Am. B 15, 1155-1159 (1998).
    [CrossRef]
  7. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
    [CrossRef]
  8. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
    [CrossRef]
  9. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
    [CrossRef] [PubMed]
  10. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
    [CrossRef]
  11. M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
    [CrossRef]
  12. K.-M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 653152-3155 (1990).
    [CrossRef] [PubMed]
  13. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001), Chap. 2.
  14. M. S. Li, S. T. Wu, and A.Yi-.G. Fuh, "Superprism phenomenon based on holographic polymer dispersed liquid crystal films," Appl. Phys. Lett. 88, 91109 (2006).
    [CrossRef]
  15. J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
    [CrossRef]
  16. G. Alagappan, X. W. Sun, P. Shum, and M. B. Yu, "Tunable superprism and polarization splitting in a liquid crystal infiltrated two-dimensional photonic crystal made of silicon oxynitride," Opt. Lett. 31, 1109-1111 (2006).
    [CrossRef] [PubMed]
  17. Y. J. Liu and X. W. Sun, "Electrically tunable two-dimensional holographic photonic crystal fabricated by a single diffractive element," Appl. Phys. Lett. 89, 171101 (2001).
    [CrossRef]
  18. W. A. Harrison, "Fermi surface in aluminum," Phys. Rev. 116, 555-561 (1959).
    [CrossRef]
  19. J. M. Ziman, Principles of the Theory of Solids, 2nd ed. (Cambridge U. Press, 1972), Chap. 3.
  20. K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
    [CrossRef]
  21. K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
    [CrossRef]
  22. C. Y. Liu and L. W. Chen, "Tunable band gap in a photonic crystal modulated by a nematic liquid crystal," Phys. Rev. B 72, 045133 (2005).
    [CrossRef]
  23. G. Alagappan, X. W. Sun, P. Shum, and M. B. Yu, "Symmetry properties of two-dimensional anisotropic photonic crystals," J. Opt. Soc. Am. A 23, 2002-2013 (2006).
    [CrossRef]
  24. G. Alagappan, X. W. Sun, P. Shum, and M. B. Yu, "Engineering the bandgap of a two-dimensional anisotropic photonic crystal," J. Opt. Soc. Am. B 23, 1478-1483 (2006).
    [CrossRef]
  25. H. Takeda and K. Yoshino, "Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals," Phys. Rev. E 67, 056607 (2003).
    [CrossRef]
  26. S. Gasiorowicz, Quantum Physics (Willey, 2003), Chap. 6.

2007 (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

2006 (4)

2005 (1)

C. Y. Liu and L. W. Chen, "Tunable band gap in a photonic crystal modulated by a nematic liquid crystal," Phys. Rev. B 72, 045133 (2005).
[CrossRef]

2004 (1)

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

2003 (3)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

H. Takeda and K. Yoshino, "Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals," Phys. Rev. E 67, 056607 (2003).
[CrossRef]

S. Gasiorowicz, Quantum Physics (Willey, 2003), Chap. 6.

2001 (2)

Y. J. Liu and X. W. Sun, "Electrically tunable two-dimensional holographic photonic crystal fabricated by a single diffractive element," Appl. Phys. Lett. 89, 171101 (2001).
[CrossRef]

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001), Chap. 2.

2000 (2)

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys. Rev. B 61, 15688-15696 (2000).
[CrossRef]

1999 (2)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

1998 (3)

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

B. D'Urso, O. Painter, J. O'Brien, T. Tombrello, A. Yariv, and A. Scherer, "Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities," J. Opt. Soc. Am. B 15, 1155-1159 (1998).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

1996 (1)

1995 (1)

J. D. Joannopoulus, R. D. Meade, and J. N. Winn, Photonic Crystals Molding The Flow of Light (Princeton, 1995), pp. 94-100.

1990 (1)

K.-M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 653152-3155 (1990).
[CrossRef] [PubMed]

1987 (2)

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

1972 (1)

J. M. Ziman, Principles of the Theory of Solids, 2nd ed. (Cambridge U. Press, 1972), Chap. 3.

1959 (1)

W. A. Harrison, "Fermi surface in aluminum," Phys. Rev. 116, 555-561 (1959).
[CrossRef]

Alagappan, G.

Atkin, D. M.

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Baumberg, J. J.

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

Birks, T. A.

Busch, K.

K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Chan, C. T.

K.-M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 653152-3155 (1990).
[CrossRef] [PubMed]

Charlton, M. D. C.

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

Chen, L. W.

C. Y. Liu and L. W. Chen, "Tunable band gap in a photonic crystal modulated by a nematic liquid crystal," Phys. Rev. B 72, 045133 (2005).
[CrossRef]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Dridi, K. H.

T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys. Rev. B 61, 15688-15696 (2000).
[CrossRef]

D'Urso, B.

Foteinopolou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Fuh, A.Yi-.G.

M. S. Li, S. T. Wu, and A.Yi-.G. Fuh, "Superprism phenomenon based on holographic polymer dispersed liquid crystal films," Appl. Phys. Lett. 88, 91109 (2006).
[CrossRef]

Gasiorowicz, S.

S. Gasiorowicz, Quantum Physics (Willey, 2003), Chap. 6.

Harrison, W. A.

W. A. Harrison, "Fermi surface in aluminum," Phys. Rev. 116, 555-561 (1959).
[CrossRef]

Ho, K.-M.

K.-M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 653152-3155 (1990).
[CrossRef] [PubMed]

Joannopoulus, J. D.

J. D. Joannopoulus, R. D. Meade, and J. N. Winn, Photonic Crystals Molding The Flow of Light (Princeton, 1995), pp. 94-100.

John, S.

K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Knight, J. C.

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Li, M. S.

M. S. Li, S. T. Wu, and A.Yi-.G. Fuh, "Superprism phenomenon based on holographic polymer dispersed liquid crystal films," Appl. Phys. Lett. 88, 91109 (2006).
[CrossRef]

Liu, C. Y.

C. Y. Liu and L. W. Chen, "Tunable band gap in a photonic crystal modulated by a nematic liquid crystal," Phys. Rev. B 72, 045133 (2005).
[CrossRef]

Liu, Y. J.

Y. J. Liu and X. W. Sun, "Electrically tunable two-dimensional holographic photonic crystal fabricated by a single diffractive element," Appl. Phys. Lett. 89, 171101 (2001).
[CrossRef]

Meade, R. D.

J. D. Joannopoulus, R. D. Meade, and J. N. Winn, Photonic Crystals Molding The Flow of Light (Princeton, 1995), pp. 94-100.

Netti, M. C.

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

Notomi, M.

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

O'Brien, J.

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

Painter, O.

Parker, G. J.

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

Perney, N. M. B.

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

Russell, P. St. J.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001), Chap. 2.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Scherer, A.

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Shum, P.

Sondergaard, T.

T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys. Rev. B 61, 15688-15696 (2000).
[CrossRef]

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

K.-M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 653152-3155 (1990).
[CrossRef] [PubMed]

Sun, X. W.

Takeda, H.

H. Takeda and K. Yoshino, "Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals," Phys. Rev. E 67, 056607 (2003).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Tombrello, T.

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Winn, J. N.

J. D. Joannopoulus, R. D. Meade, and J. N. Winn, Photonic Crystals Molding The Flow of Light (Princeton, 1995), pp. 94-100.

Wu, S. T.

M. S. Li, S. T. Wu, and A.Yi-.G. Fuh, "Superprism phenomenon based on holographic polymer dispersed liquid crystal films," Appl. Phys. Lett. 88, 91109 (2006).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

Yoshino, K.

H. Takeda and K. Yoshino, "Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals," Phys. Rev. E 67, 056607 (2003).
[CrossRef]

Yu, M. B.

Ziman, J. M.

J. M. Ziman, Principles of the Theory of Solids, 2nd ed. (Cambridge U. Press, 1972), Chap. 3.

Zoorob, M.

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

Appl. Phys. Lett. (4)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

M. S. Li, S. T. Wu, and A.Yi-.G. Fuh, "Superprism phenomenon based on holographic polymer dispersed liquid crystal films," Appl. Phys. Lett. 88, 91109 (2006).
[CrossRef]

J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob, and G. J. Parker, "Visible-wavelength super-refraction in photonic crystal superprisms," Appl. Phys. Lett. 85, 354-356 (2004).
[CrossRef]

Y. J. Liu and X. W. Sun, "Electrically tunable two-dimensional holographic photonic crystal fabricated by a single diffractive element," Appl. Phys. Lett. 89, 171101 (2001).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

W. A. Harrison, "Fermi surface in aluminum," Phys. Rev. 116, 555-561 (1959).
[CrossRef]

Phys. Rev. B (4)

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys. Rev. B 61, 15688-15696 (2000).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096 (1998).
[CrossRef]

C. Y. Liu and L. W. Chen, "Tunable band gap in a photonic crystal modulated by a nematic liquid crystal," Phys. Rev. B 72, 045133 (2005).
[CrossRef]

Phys. Rev. E (2)

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

H. Takeda and K. Yoshino, "Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals," Phys. Rev. E 67, 056607 (2003).
[CrossRef]

Phys. Rev. Lett. (5)

K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, "Subwavelength resolution in a two-dimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

K.-M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 653152-3155 (1990).
[CrossRef] [PubMed]

Other (4)

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001), Chap. 2.

J. M. Ziman, Principles of the Theory of Solids, 2nd ed. (Cambridge U. Press, 1972), Chap. 3.

J. D. Joannopoulus, R. D. Meade, and J. N. Winn, Photonic Crystals Molding The Flow of Light (Princeton, 1995), pp. 94-100.

S. Gasiorowicz, Quantum Physics (Willey, 2003), Chap. 6.

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Figures (5)

Fig. 1
Fig. 1

EFS constructions of a square lattice PC with f = 0.4 , n a = 1.6 and n b n a . Band indices (online): band 2, blue; band 3, red; band 4, pink; band 5, yellow. The thin lines indicate boundaries of the square lattice BZ: (a) EFS construction with the band index for ω = 0.5 , (b) EFS for ω = 0.5 , (c) EFS for ω = 0.36 , (d) EFS for ω = 0.67 .

Fig. 2
Fig. 2

EFS constructions (dotted black circular lines) and the numerically calculated EFS (color) of a square lattice PC with f = 0.4 n a = 1.6 and ω = 0.5 . Band indices (online): band 3, red; band 4, pink; band 5, yellow. The thin lines indicate boundaries of the square lattice BZ: (a) EFS for n b = 2.0 , (b) EFS n b = 2.6 .

Fig. 3
Fig. 3

EFS constructions for H-polarization (dotted black elliptical lines) and the numerically calculated EFS (color) of a square lattice PC with anisotropic materials ( f = 0.2 , n a = 1.8 , ϵ b 1 = 1.6 , ϵ b 2 = 2.0 , ω = 0.5 ) : band 2, blue; band 3, red; band 4, pink. (a) EFS construction, (b) Numerically calculated EFS with band index.

Fig. 4
Fig. 4

EFS approximations using two plane waves for a square lattice PC with n a = 1.6 , n b = 1.8 , and f = 0.2 . The numerically calculated EFS [thin (black) lines] and the approximated EFS [thick (blue) lines] for frequencies in the (a) first band and (b) second band.

Fig. 5
Fig. 5

Merit of two-plane waves: EFS approximation for a square lattice PC with n a = 1.6 , f = 0.2 . EFS ( ω = 0.31 ) obtained from the one-plane-wave approximation (dashed gray line), two-plane-wave approximation [thick blue (online) line], and numerical calculation [thin red (online) line] for (a) n b = 1.8 , (b) n b = 2.0 .

Equations (21)

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M A = ( 2 π ω a ) 2 A ,
Ω = i j A G i G i M G j G j A ,
Ω = i j A j A i * M i j .
M i j = k i β ̃ r ( i j ) k j ,
M i j = k i k j β ( i j ) ,
ϵ ̃ m = ϵ ̃ a f + ( 1 f ) ϵ ̃ b .
k i ϵ ̃ m 1 k i = Ω ,
Ω = k i Q T ϵ ̃ m p 1 Q k i = Q k i ϵ ̃ m p 1 Q k i ,
ϵ ̃ m p 1 = ( 1 { ϵ a f + ϵ b 1 [ 1 f ] } 0 0 1 { ϵ a f + ϵ b 2 [ 1 f ] } ) ,
( k x G i x ) 2 ϵ a f + ϵ b 1 [ 1 f ] + ( k y G i y ) 2 ϵ a f + ϵ b 2 [ 1 f ] = Ω .
Ω = A i M i , i A i + A i M i , i + 1 A i + 1 + A i + 1 M i + 1 , i A i + A i + 1 M i + 1 , i + 1 A i + 1 ,
M = [ M 11 Ω M 12 M 21 M 22 Ω ] .
2 Ω = β 0 ( k 0 2 + k 1 2 ) ± β 0 2 ( k 0 2 k 1 2 ) 2 + ( 2 β 1 k 0 k 1 ) 2 ,
β 0 ( k 0 2 + k 1 2 ) = 2 β 0 [ ( k x + g 2 ) 2 + k y 2 + g 2 4 ] ,
β 0 2 ( k 0 2 k 1 2 ) 2 = [ 2 β 0 g ( k x + g 2 ) ] 2 ,
( 2 β 1 k 0 k 1 ) 2 = 4 β 1 2 [ ( k x + g 2 ) 2 + k y 2 + g 2 4 ] 2 [ 2 β 1 ( k x + g 2 ) ] 2 .
Ω = β 0 ( k X 2 + k y 2 + g 2 4 ) ± ( β 0 k X g ) 2 + β 1 2 ( k X 2 + k y 2 + g 2 4 ) 2 ( β 1 k X g ) 2 .
[ Ω 1 g 2 4 ( β 0 β 1 ) ] = k X 2 ( β 0 β 1 2 β 0 2 β 1 ) + k y 2 ( β 0 β 1 ) ,
[ Ω 2 g 2 4 ( β 0 + β 1 ) ] = k X 2 ( β 0 + β 1 + 2 β 0 2 β 1 ) + k y 2 ( β 0 + β 1 ) ,
β 1 β 0 g ( k y 2 + g 2 4 ) < k X < β 1 β 0 g ( k y 2 + g 2 4 ) .
k y > g 2 β 1 β 0 2 β 1 2 .

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