Abstract

Optimal design of a two-dimensional photonic crystal with a square lattice of air holes in GaAs is considered. It is shown how a maximum complete two-dimensional band gap is obtained by optimally connecting the dielectric rods with veins. The complete two-dimensional bandgap of our optimal design reaches Δω=0.0762(2πc/a)(a is the lattice constant).

© 2000 Optical Society of America

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. J. D. Joannopoulos, R. D. Mead, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  3. C. M. Soukoulis, ed., Photonic Band Gaps and Localization, Proceedings of the NATO Advanced Research Workshops (Plenum, New York, 1993).
  4. J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature (London) 386, 143–149 (1997).
    [CrossRef]
  5. 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]
  6. D. L. Bullock, C. Shih, and R. S. Margulies, “Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,” J. Opt. Soc. Am. B 10, 399–403 (1993).
    [CrossRef]
  7. U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm,” Appl. Phys. Lett. 68, 747–749 (1996).
    [CrossRef]
  8. J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
    [CrossRef]
  9. S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. 41, 385–393 (1994).
    [CrossRef]
  10. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 81, 2574–2577 (1998).
    [CrossRef]
  11. C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
    [CrossRef] [PubMed]
  12. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic bandgap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  13. K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
    [CrossRef] [PubMed]
  14. S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239–1242 (1990).
    [CrossRef] [PubMed]
  15. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  16. M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199–204 (1991).
    [CrossRef]
  17. P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
    [CrossRef]
  18. E. Polack, Computational Methods in Optimization, a Unified Approach (Academic, New York, 1971).
  19. S. He, P. Fuks, and G. Larson, “An optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
    [CrossRef]

1998

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

1997

J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

1996

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]

U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

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

S. He, P. Fuks, and G. Larson, “An optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

1994

S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. 41, 385–393 (1994).
[CrossRef]

1993

J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
[CrossRef]

D. L. Bullock, C. Shih, and R. S. Margulies, “Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,” J. Opt. Soc. Am. B 10, 399–403 (1993).
[CrossRef]

1992

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[CrossRef]

1991

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. Commun. 80, 199–204 (1991).
[CrossRef]

1990

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic bandgap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239–1242 (1990).
[CrossRef] [PubMed]

1987

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

Anderson, C. M.

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

Arjavalingam, G.

S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. 41, 385–393 (1994).
[CrossRef]

Atkin, D. M.

Birks, T. A.

Brennan, T. M.

J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
[CrossRef]

Bullock, D. L.

Busch, K.

U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic bandgap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Fan, S.

J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Fuks, P.

S. He, P. Fuks, and G. Larson, “An optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

Giapis, K. P.

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

Gourley, P. L.

J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
[CrossRef]

Grüning, U.

U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Gu, B. Y.

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

Hammons, B. E.

J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
[CrossRef]

He, S.

S. He, P. Fuks, and G. Larson, “An optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic bandgap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

J. Russell, P. St.

Joannopoulos, J. D.

J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Knight, J. C.

Larson, G.

S. He, P. Fuks, and G. Larson, “An optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

Lehmann, V.

U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Leung, K. M.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Li, Z. Y.

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

Lin, S. Y.

S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. 41, 385–393 (1994).
[CrossRef]

Liu, Y. F.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Maradudin, A. A.

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. Commun. 80, 199–204 (1991).
[CrossRef]

Margulies, R. S.

Ottow, S.

U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Piché, M.

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[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. Commun. 80, 199–204 (1991).
[CrossRef]

Robertson, W. M.

S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. 41, 385–393 (1994).
[CrossRef]

Salehpour, M. R.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239–1242 (1990).
[CrossRef] [PubMed]

Satpathy, S.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239–1242 (1990).
[CrossRef] [PubMed]

Shambrook, A.

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

Sheng, P.

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

Shih, C.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic bandgap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Vawter, G. A.

J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
[CrossRef]

Villeneuve, P.

J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Villeneuve, P. R.

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[CrossRef]

Wendt, J. R.

J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
[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.

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

Zhang, Z.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239–1242 (1990).
[CrossRef] [PubMed]

Appl. Phys. Lett.

U. Grüning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

IEEE Trans. Antennas Propag.

S. He, P. Fuks, and G. Larson, “An optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

J. Mod. Opt.

S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. 41, 385–393 (1994).
[CrossRef]

J. Opt. Soc. Am. B

J. Vac. Sci. Technol. B

J. R. Wendt, G. A. Vawter, P. L. Gourley, T. M. Brennan, and B. E. Hammons, “Nanofabrication of photonic lattice structures in GaAs/AlGaAs,” J. Vac. Sci. Technol. B 11, 2637–2640 (1993).
[CrossRef]

Nature (London)

J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Phys. Rev. B

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[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]

Phys. Rev. Lett.

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

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

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

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic bandgap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239–1242 (1990).
[CrossRef] [PubMed]

Other

J. D. Joannopoulos, R. D. Mead, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

C. M. Soukoulis, ed., Photonic Band Gaps and Localization, Proceedings of the NATO Advanced Research Workshops (Plenum, New York, 1993).

E. Polack, Computational Methods in Optimization, a Unified Approach (Academic, New York, 1971).

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Figures (7)

Fig. 1
Fig. 1

Evolution of the design for a photonic crystal with a large complete two-dimensional bandgap. The unit cell is indicated by the dashed frame. (a) Dielectric columns in air, (b) dielectric veins in air, (c) dielectric columns connected by veins, (d) another view of (c) with a single air hole in each cell.

Fig. 2
Fig. 2

Band structures for the GaAs columns in air. The radius for the GaAs rods is R=0.310a.

Fig. 3
Fig. 3

Band structures for the GaAs veins in air. The half-vein width is d=0.12a.

Fig. 4
Fig. 4

Band structures for the designed photonic crystal shown in Fig. 1(c). The radius for the GaAs rods is R=0.310a, and the half-vein width is d=0.038a. The complete two-dimensional bandgap is approximately Δω=0.0685(2πc/a).

Fig. 5
Fig. 5

Gap map as the half-vein width d varies. The radius for the GaAs rods is fixed to R=0.310a.

Fig. 6
Fig. 6

Gap map as the radius R for the GaAs rods varies. The half-vein width is fixed to d=0.038a.

Fig. 7
Fig. 7

Optimal design (solid curve) obtained by the conjugate gradient method [the corresponding complete two-dimensional bandgap is approximately Δω=0.0762(2πc/a)]. The dotted curve is our original design (corresponding to the band structures shown in Fig. 4).

Equations (1)

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r(θ)=r0(θ)+A0+n=1NAn cos(4nθ).

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