Abstract

We investigated the transmission spectra and the Bragg-reflection spectra of a two-dimensional photonic crystal composed of a triangular array of circular air rods formed in PbO glass, for which the laser oscillation peculiar to the two-dimensional photonic band structure was observed recently. The sample parameters, i.e., the lattice constant, the radius of the air rods, and the dielectric constant of the host PbO glass, were evaluated from the observation angle of the Bragg reflection and by comparison of observed with calculated band gaps. The transmission spectra and the Bragg-reflection spectra were calculated with the plane-wave expansion method, and a good agreement with both the dispersion relation and the observed Bragg-reflection spectra was shown.

© 1999 Optical Society of America

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References

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  1. For fundamental ideas and properties of photonic crystals see J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University, Princeton, N.J. 1995); C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993); C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  3. G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269 (1988).
    [CrossRef] [PubMed]
  4. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418 (1990); “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43, 12, 772 (1991).
    [CrossRef] [PubMed]
  5. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
    [CrossRef] [PubMed]
  6. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
    [CrossRef] [PubMed]
  7. D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947 (1987).
    [CrossRef]
  8. S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168 (1993).
    [CrossRef] [PubMed]
  9. S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50, 1764 (1994); “Localization of superradiance near a photonic band gap,” Phys. Rev. Lett. 74, 3419 (1995).
    [CrossRef] [PubMed]
  10. K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732 (1996); “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742 (1996).
    [CrossRef]
  11. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896 (1994).
    [CrossRef]
  12. K. Sakoda, “Enhanced stimulated emission in a two-dimensional photonic crystal,” in Proceedings of the 1998 International Conference on Application of Photonic Technology (Institute of Electrical and Electronics Engineers, Piscataway, N.J., to be published).
  13. M. Sasada, A. Yamanaka, K. Sakoda, K. Inoue, and J. W. Haus, “Laser oscillation from dye molecules in a two-dimensional photonic crystal,” in Proceedings of the Conference on Lasers and Electro-Optics/Pacific Rim (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), p. 42.
  14. K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “Laser action characteristic of a two-dimensional photonic lattice,” in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 47.
  15. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565 (1991).
    [CrossRef]
  16. K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
    [CrossRef]
  17. K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
    [CrossRef]
  18. K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672 (1995); “Transmittance and Bragg reflectivity of a two-dimensional photonic lattices,” Phys. Rev. B 52, 8992 (1995); “Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice,” J. Opt. Soc. Am. B JOBPDE 14, 1961 (1997).
    [CrossRef]
  19. K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
    [CrossRef]
  20. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023 (1992); “Measurement of the photon dispersion relation in two-dimensional ordered dielectric arrays,” J. Opt. Soc. Am. B 10, 322 (1993).
    [CrossRef] [PubMed]
  21. K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves III: group-theoretical treatment,” J. Phys. Soc. Jpn. 65, 2670 (1996).
    [CrossRef]
  22. K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15, 345 (1997).
    [CrossRef]

1997 (1)

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15, 345 (1997).
[CrossRef]

1996 (2)

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves III: group-theoretical treatment,” J. Phys. Soc. Jpn. 65, 2670 (1996).
[CrossRef]

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

1995 (1)

K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
[CrossRef]

1994 (2)

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896 (1994).
[CrossRef]

1993 (1)

S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168 (1993).
[CrossRef] [PubMed]

1991 (3)

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

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

1988 (1)

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269 (1988).
[CrossRef] [PubMed]

1987 (2)

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

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947 (1987).
[CrossRef]

Aközbek, N.

S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168 (1993).
[CrossRef] [PubMed]

Bloemer, M. J.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896 (1994).
[CrossRef]

Bowden, C. M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896 (1994).
[CrossRef]

Brommer, K. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Dalichaouch, R.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Dowling, J. P.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896 (1994).
[CrossRef]

Fukushima, T.

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

Genack, A. Z.

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269 (1988).
[CrossRef] [PubMed]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Haus, J. W.

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

Hayashi, M.

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

Inoue, K.

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

Joannopoulos, J. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

John, S.

S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168 (1993).
[CrossRef] [PubMed]

Kurizki, G.

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269 (1988).
[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 (1991).
[CrossRef]

McCall, S. L.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Meade, R. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Mills, D. L.

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947 (1987).
[CrossRef]

Ohtaka, K.

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves III: group-theoretical treatment,” J. Phys. Soc. Jpn. 65, 2670 (1996).
[CrossRef]

Platzman, P. M.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Plihal, M.

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

Rappe, A. M.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Sakoda, K.

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15, 345 (1997).
[CrossRef]

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
[CrossRef]

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

Scalora, M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896 (1994).
[CrossRef]

Schultz, S.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Smith, D.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Tanabe, Y.

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves III: group-theoretical treatment,” J. Phys. Soc. Jpn. 65, 2670 (1996).
[CrossRef]

Trullinger, S. E.

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947 (1987).
[CrossRef]

Wada, M.

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

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

Yamanaka, A.

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

J. Appl. Phys. (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896 (1994).
[CrossRef]

J. Phys. Soc. Jpn. (1)

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves III: group-theoretical treatment,” J. Phys. Soc. Jpn. 65, 2670 (1996).
[CrossRef]

Jpn. J. Appl. Phys., Part 2 (1)

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of two-dimensional photonic band structure with near-infrared band gap,” Jpn. J. Appl. Phys., Part 2 33, L1463 (1994).
[CrossRef]

Phys. Rev. B (5)

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

K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattice as revealed by transmittance measurement,” Phys. Rev. B 53, 1010 (1996).
[CrossRef]

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947 (1987).
[CrossRef]

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15, 345 (1997).
[CrossRef]

K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
[CrossRef]

Phys. Rev. Lett. (5)

S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168 (1993).
[CrossRef] [PubMed]

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

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269 (1988).
[CrossRef] [PubMed]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Other (9)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418 (1990); “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43, 12, 772 (1991).
[CrossRef] [PubMed]

S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50, 1764 (1994); “Localization of superradiance near a photonic band gap,” Phys. Rev. Lett. 74, 3419 (1995).
[CrossRef] [PubMed]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732 (1996); “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742 (1996).
[CrossRef]

K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672 (1995); “Transmittance and Bragg reflectivity of a two-dimensional photonic lattices,” Phys. Rev. B 52, 8992 (1995); “Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice,” J. Opt. Soc. Am. B JOBPDE 14, 1961 (1997).
[CrossRef]

K. Sakoda, “Enhanced stimulated emission in a two-dimensional photonic crystal,” in Proceedings of the 1998 International Conference on Application of Photonic Technology (Institute of Electrical and Electronics Engineers, Piscataway, N.J., to be published).

M. Sasada, A. Yamanaka, K. Sakoda, K. Inoue, and J. W. Haus, “Laser oscillation from dye molecules in a two-dimensional photonic crystal,” in Proceedings of the Conference on Lasers and Electro-Optics/Pacific Rim (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), p. 42.

K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “Laser action characteristic of a two-dimensional photonic lattice,” in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 47.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023 (1992); “Measurement of the photon dispersion relation in two-dimensional ordered dielectric arrays,” J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

For fundamental ideas and properties of photonic crystals see J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University, Princeton, N.J. 1995); C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993); C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer Academic, Dordrecht, The Netherlands, 1996).

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Figures (6)

Fig. 1
Fig. 1

Top view of the geometry for the numerical calculation of the transmission spectra and the Bragg-reflection spectra by means of the plane-wave expansion method. εa and εb stand for the dielectric constants of air and PbO glass. a, R, and N denote the lattice constant, the radius of the cylindrical air-holes, and the number of the lattice layers, respectively. The intensity of the transmitted wave with a wave vector kt(0) (=ki) and the reflected waves with wave vectors kr(0) (=-ki) and kr(±1) were calculated, where ki is the wave vector of the incident light.

Fig. 2
Fig. 2

Observation-angle dependence of the reflected light intensity measured at 710 nm with s-polarized (E-polarized) incident light. In addition to the specular reflection at 0 deg, the Bragg reflection of the first order is clearly observed at ±37.9 deg.

Fig. 3
Fig. 3

Transmittance (right-hand side) and dispersion relation (left-hand side) for E polarization in the ΓX direction. The ordinate is the normalized frequency. The following values were assumed for the numerical calculation: a=1.15 µm, R =0.486 µm, εa=1.0 (air), εb=2.72 (PbO glass), and N =16. On the left-hand side, solid curves represent symmetric modes and dashed curves represent antisymmetric modes. The latter cannot be excited by the incident plane wave because of the mismatching of their spatial symmetry, and so they do not contribute to the light transmission. Opaque frequency regions are clearly observed in the transmission spectrum where no symmetric mode exists.

Fig. 4
Fig. 4

Transmittance (right-hand side) and dispersion relation (left-hand side) for H polarization in the ΓX direction. The ordinate is the normalized frequency. The same parameters as for Fig. 3 were assumed for the numerical calculation. On the left-hand side, solid and dashed curves represent symmetric and antisymmetric modes, respectively.

Fig. 5
Fig. 5

Observed (circles) and calculated (solid curve) reflectivity for the zeroth-order Bragg reflection. The abscissa is the normalized frequency.

Fig. 6
Fig. 6

Observed (circles) and the calculated (solid curve) reflectivity for the first-order Bragg reflection. The abscissa is the normalized frequency.

Tables (1)

Tables Icon

Table 1 Comparison of the Observed Bandgap Frequencies at the X Point with the Calculated Onesa

Equations (7)

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

kr(n)=kt(n)=ki+2nπa,
|kr(n)|2=|kt(n)|2=|ki|2,
1ε(x, y)=n=-m=-κnm expi2nπax+2mπ3Nay.
κ0,0=fεa+1-fεb,
κn,lN=f1εa-1εb[1+(-1)n+l] J1(Gn,lNR)Gn,lNR
(forn0orl0),
Gnm=2nπa2+2mπ3Na21/2.

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