Abstract

Using the block-iterative frequency domain method and the non-orthogonal FDTD method, the photonic band gap (PBG) and spectral properties are investigated for a new class of two-dimensional (2-D) trigonal structures with an approximately circular or hexagonal “atom” shape formed by holographic lithography. Calculations of band structures as a function of the intensity threshold show that the PBG of 2-D titania arrays opens only for TM polarization, and directional PBG can open for TE and TM polarization simultaneously. In addition, up to four sizeable full PBGs can open for an inverted GaAs triangular structure.

© 2003 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  5. B. D’Urso, O. Painter, J.D. 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]
  6. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2003 (1)

2002 (1)

2001 (4)

A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[Crossref]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
[Crossref] [PubMed]

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[Crossref]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26, 1858–1860 (2001).
[Crossref]

2000 (2)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

1999 (1)

X. H. Wang, B. Y. Gu, Z. Y. Li, and G. Z. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B 60, 11417–11421 (1999).
[Crossref]

1998 (3)

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

B. D’Urso, O. Painter, J.D. 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]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

1997 (2)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4 mm photonic crystals”, Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

1996 (3)

C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[Crossref] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[Crossref] [PubMed]

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134–7142 (1996).
[Crossref]

1994 (2)

R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. 41, 295–310 (1994).
[Crossref]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

1993 (1)

1991 (2)

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991).
[Crossref]

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

1987 (2)

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

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Anderson, C. M.

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4 mm photonic crystals”, Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[Crossref] [PubMed]

Atkin, D. M.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Berger, V.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Bertho, D.

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134–7142 (1996).
[Crossref]

Birks, T. A.

Brent, Richard

Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).

Bullock, D. L.

Cai, L. Z.

Cassagne, D.

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134–7142 (1996).
[Crossref]

Cheng, B.

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

Costard, E.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

D’Urso, B.

B. D’Urso, O. Painter, J.D. 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]

Diviliansky, Ivan B.

A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[Crossref]

Gauthier-Lafaye, O.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Gerard, J. M.

R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. 41, 295–310 (1994).
[Crossref]

Giapis, K. P.

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4 mm photonic crystals”, Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[Crossref] [PubMed]

Gu, B. Y.

X. H. Wang, B. Y. Gu, Z. Y. Li, and G. Z. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B 60, 11417–11421 (1999).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Jin, C.

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

Joannopoulos, J. D.

John, S.

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Johnson, S. G.

Jouanin, C.

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134–7142 (1996).
[Crossref]

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[Crossref]

Khoo, I. C.

A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[Crossref]

Knight, J. C.

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[Crossref]

Li, L. M.

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

Li, Z. Y.

X. H. Wang, B. Y. Gu, Z. Y. Li, and G. Z. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B 60, 11417–11421 (1999).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Liu, Q.

Man, B.

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

Maradudin, A. A.

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991).
[Crossref]

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

Margulies, R. S.

Marzin, J. Y.

R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. 41, 295–310 (1994).
[Crossref]

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[Crossref]

Mayer, T. S.

A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[Crossref]

Misawa, H.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[Crossref]

O’Brien, J.D.

B. D’Urso, O. Painter, J.D. 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]

Padjen, R.

R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. 41, 295–310 (1994).
[Crossref]

Painter, O.

B. D’Urso, O. Painter, J.D. 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]

Pendry, J. B.

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991).
[Crossref]

Russell, P. St. J.

Scherer, A.

B. D’Urso, O. Painter, J.D. 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]

Shambrook, A.

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991).
[Crossref]

Sheng, P.

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991).
[Crossref]

Shih, C.

Shishido, A.

A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[Crossref]

Tombrello, T.

B. D’Urso, O. Painter, J.D. 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]

Wang, X. H.

X. H. Wang, B. Y. Gu, Z. Y. Li, and G. Z. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B 60, 11417–11421 (1999).
[Crossref]

Wang, Y. R.

Ward, A. J.

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

Yablonovitch, E.

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

Yang, G. Z.

X. H. Wang, B. Y. Gu, Z. Y. Li, and G. Z. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B 60, 11417–11421 (1999).
[Crossref]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

Yang, X. L.

Yariv, A.

B. D’Urso, O. Painter, J.D. 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]

Zhang, D.

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

Zhang, X.

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

Zhang, Z.Q.

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

Appl. Phys. Lett. (2)

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[Crossref]

A. Shishido, Ivan B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[Crossref]

J. Appl. Phys. (1)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

J. Mod. Opt. (1)

R. Padjen, J. M. Gerard, and J. Y. Marzin, “Analysis of the filling pattern dependence of the photonic bandgap for two-dimensional systems,” J. Mod. Opt. 41, 295–310 (1994).
[Crossref]

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

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

B. D’Urso, O. Painter, J.D. 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]

Nature (1)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Opt. Comm. (1)

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199–204 (1991).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. B (6)

X. H. Wang, B. Y. Gu, Z. Y. Li, and G. Z. Yang, “Large absolute photonic band gaps created by rotating noncircular rods in two-dimensional lattices,” Phys. Rev. B 60, 11417–11421 (1999).
[Crossref]

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4 mm photonic crystals”, Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[Crossref]

X. Zhang, Z.Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, “Enlarging a photonic band gap by using insertion,” Phys. Rev. B 61, 1892–1897 (2000).
[Crossref]

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134–7142 (1996).
[Crossref]

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

Phys. Rev. Lett. (4)

C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[Crossref] [PubMed]

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
[Crossref]

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

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Other (1)

Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).

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Figures (5)

Fig. 1.
Fig. 1.

(a) 2-D triangular photonic lattice fabricated by the interference technique of three noncoplanar beam and the first Brillouin zone with the symmetry points indicated; (b) and (c), dotted lines (I) present the relation between the intensity threshold and the FR of dielectric, and solid lines (II) present the derivative of curves (I), where (b) is for titania and (c) is for GaAs.

Fig. 2.
Fig. 2.

(a) TM gap map for the 2-D triangular titania arrays; (b) TM photonic band structure for It =3.0.

Fig. 3.
Fig. 3.

(a) Gap map of directional PBG for 2-D titania arrays; (b) Gap map of full PBG for the inverted GaAs structure, where blue area is for TM polarization, red is for TE polarization and yellow area is for both.

Fig. 4.
Fig. 4.

The directional photonic band diagrams (a and c) and calculated transmission spectra (b and d) for TE (red line) and TM (blue line) polarizations respectively, where (a) and (b) are for It =3.0, and (c) and (d) are for It =4.6.

Fig. 5.
Fig. 5.

Photonic band structures when the intensity threshold is 1.6 (a) and 2.0 (b) for TE (red line) and TM (blue line) polarizations, respectively.

Equations (1)

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I 0 = 3 + cos [ 2 π 3 a ( 2 y ) ] + cos [ 2 π 3 a ( 3 3 x + y ) ] + cos [ 2 π 3 a ( 3 3 x y ) ] ,

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