Abstract

We present a three-dimensional finite-difference time-domain analysis of localized defect modes in an optically thin dielectric slab that is patterned with a two-dimensional array of air holes. The symmetry, quality factor, and radiation pattern of the defect modes and their dependence on the slab thickness are investigated.

© 1999 Optical Society of America

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  1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  2. D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
    [CrossRef]
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  4. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  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, 2017–2020 (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, 3380–3383 (1991).
    [CrossRef] [PubMed]
  7. B. P. V. der Gaag and A. Scherer, Appl. Phys. Lett. 56, 481–483 (1989).
    [CrossRef]
  8. C. C. Cheng, A. Scherer, V. Arbet-Engels, and E. Yablonovitch, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4119 (1996).
    [CrossRef]
  9. T. Krauss, Y. P. Song, S. Thoms, C. D. W. Wilkinson, and R. M. D. L. Rue, “Fabrication of 2-D photonic bandgap structures in GaAs/AlGaAs,” Electron. Lett. 30, 1444–1446 (1994).
    [CrossRef]
  10. G. Feiertag, W. Ehrfeld, H. Freimuth, H. Kolle, H. Lehr, M. Schmidt, M. M. Sigalas, C. M. Soukoulis, G. Kiriakidis, T. Pederson, J. Kuhl, and W. Koenig, “Fabrication of photonic crystals by deep x-ray lithography,” Appl. Phys. Lett. 71, 1441–1443 (1997).
    [CrossRef]
  11. K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattices as revealed by transmittance measurement,” Phys. Rev. B 53, 1010–1013 (1996).
    [CrossRef]
  12. J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, and L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. QE-27, 1332–1346 (1996).
  13. A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometre radius disk laser,” Electron. Lett. 29, 1666–1667 (1993).
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  14. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
    [CrossRef]
  15. D. Y. Chu and S.-T. Ho, “Spontaneous emission from excitons in cylindrical dielectric waveguides and the spontaneous-emission factor of microcavity ring lasers,” J. Opt. Soc. Am. B 10, 381–390 (1993).
    [CrossRef]
  16. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
    [CrossRef]
  17. J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
    [CrossRef]
  18. T. F. Krauss, B. Vögele, C. R. Stanley, and R. M. D. L. Rue, “Waveguide microcavity based on photonic microstructures,” IEEE Photonics Technol. Lett. 9, 176–178 (1997).
    [CrossRef]
  19. T. F. Krauss, R. M. D. L. Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
    [CrossRef]
  20. J. O’Brien, O. Painter, C. C. Cheng, R. Lee, A. Scherer, and A. Yariv, “Lasers incorporating 2D photonic bandgap mirrors,” Electron. Lett. 32, 2243–2244 (1996).
    [CrossRef]
  21. T. Baba and T. Matsuzaki, “Fabrication and photoluminescence of GaInAsP/InP 2D photonic crystals,” Jpn. J. Appl. Phys., Part 2 35, 1348–1352 (1996).
    [CrossRef]
  22. T. Hamano, H. Hirayama, and Y. Aoyyagi, “Optical characterization of GaAs 2D photonic bandgap crystal fabricated by selective MOVPE,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 528–529.
  23. T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
    [CrossRef]
  24. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [CrossRef]
  25. H. Yokoyama, “Physics and device application of optical microcavities,” Science 256, 66–70 (1992).
    [CrossRef] [PubMed]
  26. I. Schnitzer, E. Yablonovitch, A. Scherer, and T. J. Gmitter, in Photonic Band Gaps and Localization (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 369–378.
  27. K. S. Yee, “Numerical solution of boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  28. G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. QE-27, 2386–2396 (1991).
    [CrossRef]
  29. D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
    [CrossRef] [PubMed]
  30. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
  31. O. Painter, R. Lee, A. Yariv, A. Scherer, and J. O’Brien, “Photonic bandgap membrane microresonator,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 221–223.
  32. Y. Zou, J. S. Osinski, P. Grodzinski, P. D. Dapkus, W. Rideout, W. F. Sharfim, and F. D. Crawford, “Experimental verification of strain benefits in 1.5 μm semiconductor lasers by carrier lifetime and gain measurements,” IEEE Photonics Technol. Lett. 4, 1315–1318 (1992).
    [CrossRef]
  33. S. Adachi, “Material parameters of In1−xGaxAsyP1−y and related binaries,” J. Appl. Phys. 53, 8775–8792 (1982).
    [CrossRef]
  34. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  35. D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
    [CrossRef]
  36. C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16, 635–16, 642 (1995).
    [CrossRef]
  37. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981).
    [CrossRef]
  38. P. S. J. Russell, D. M. Atkin, and T. A. Birks, in Microcavi ties and Photonic Bandgaps (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 203–218.
  39. K. Chamberlain and L. Gordon, “Modeling good conductors using the finite-difference, time-domain technique,” IEEE Trans. Electromagn. Compat. 37, 210–216 (1995).
    [CrossRef]
  40. B. D’Urso, O. Painter, J. O’Brien, T. Tombrello, A. Scherer, and A. Yariv, “Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,” J. Opt. Soc. Am. B 15, 1155–1159 (1998).
    [CrossRef]
  41. D. H. Choi and W. J. R. Hoefer, “The finite-difference-time-domain method and its application to eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 34, 1464–1469 (1986).
    [CrossRef]
  42. K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic crystals,” Phys. Rev. B 52, 7982–7986 (1995).
    [CrossRef]
  43. M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964).
  44. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  45. O. Painter, J. Vučković, and A. Scherer, in an unpublished study, analyze similar microcavities, as presented in this paper, although with different bottom substrates of the waveguide and with increasing number of photonic crystal layers. Calculations of QT, Q, and Q versus frequency of the defect mode with seven layers of photonic crystal shows a smooth peak for all the Q’s, with none of the structure present for only three layers.

1998 (1)

1997 (4)

G. Feiertag, W. Ehrfeld, H. Freimuth, H. Kolle, H. Lehr, M. Schmidt, M. M. Sigalas, C. M. Soukoulis, G. Kiriakidis, T. Pederson, J. Kuhl, and W. Koenig, “Fabrication of photonic crystals by deep x-ray lithography,” Appl. Phys. Lett. 71, 1441–1443 (1997).
[CrossRef]

T. F. Krauss, B. Vögele, C. R. Stanley, and R. M. D. L. Rue, “Waveguide microcavity based on photonic microstructures,” IEEE Photonics Technol. Lett. 9, 176–178 (1997).
[CrossRef]

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
[CrossRef]

1996 (9)

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
[CrossRef]

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

T. F. Krauss, R. M. D. L. Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

J. O’Brien, O. Painter, C. C. Cheng, R. Lee, A. Scherer, and A. Yariv, “Lasers incorporating 2D photonic bandgap mirrors,” Electron. Lett. 32, 2243–2244 (1996).
[CrossRef]

T. Baba and T. Matsuzaki, “Fabrication and photoluminescence of GaInAsP/InP 2D photonic crystals,” Jpn. J. Appl. Phys., Part 2 35, 1348–1352 (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 lattices as revealed by transmittance measurement,” Phys. Rev. B 53, 1010–1013 (1996).
[CrossRef]

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, and L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. QE-27, 1332–1346 (1996).

C. C. Cheng, A. Scherer, V. Arbet-Engels, and E. Yablonovitch, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4119 (1996).
[CrossRef]

1995 (3)

K. Chamberlain and L. Gordon, “Modeling good conductors using the finite-difference, time-domain technique,” IEEE Trans. Electromagn. Compat. 37, 210–216 (1995).
[CrossRef]

K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic crystals,” Phys. Rev. B 52, 7982–7986 (1995).
[CrossRef]

C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16, 635–16, 642 (1995).
[CrossRef]

1994 (1)

T. Krauss, Y. P. Song, S. Thoms, C. D. W. Wilkinson, and R. M. D. L. Rue, “Fabrication of 2-D photonic bandgap structures in GaAs/AlGaAs,” Electron. Lett. 30, 1444–1446 (1994).
[CrossRef]

1993 (2)

D. Y. Chu and S.-T. Ho, “Spontaneous emission from excitons in cylindrical dielectric waveguides and the spontaneous-emission factor of microcavity ring lasers,” J. Opt. Soc. Am. B 10, 381–390 (1993).
[CrossRef]

A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometre radius disk laser,” Electron. Lett. 29, 1666–1667 (1993).
[CrossRef]

1992 (3)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Y. Zou, J. S. Osinski, P. Grodzinski, P. D. Dapkus, W. Rideout, W. F. Sharfim, and F. D. Crawford, “Experimental verification of strain benefits in 1.5 μm semiconductor lasers by carrier lifetime and gain measurements,” IEEE Photonics Technol. Lett. 4, 1315–1318 (1992).
[CrossRef]

H. Yokoyama, “Physics and device application of optical microcavities,” Science 256, 66–70 (1992).
[CrossRef] [PubMed]

1991 (4)

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. QE-27, 2386–2396 (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, 2017–2020 (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, 3380–3383 (1991).
[CrossRef] [PubMed]

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

1989 (1)

B. P. V. der Gaag and A. Scherer, Appl. Phys. Lett. 56, 481–483 (1989).
[CrossRef]

1987 (2)

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

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

1986 (1)

D. H. Choi and W. J. R. Hoefer, “The finite-difference-time-domain method and its application to eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 34, 1464–1469 (1986).
[CrossRef]

1982 (1)

S. Adachi, “Material parameters of In1−xGaxAsyP1−y and related binaries,” J. Appl. Phys. 53, 8775–8792 (1982).
[CrossRef]

1981 (2)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981).
[CrossRef]

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Adachi, S.

S. Adachi, “Material parameters of In1−xGaxAsyP1−y and related binaries,” J. Appl. Phys. 53, 8775–8792 (1982).
[CrossRef]

Arbet-Engels, V.

C. C. Cheng, A. Scherer, V. Arbet-Engels, and E. Yablonovitch, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4119 (1996).
[CrossRef]

Atkin, D. M.

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

Baba, T.

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
[CrossRef]

T. Baba and T. Matsuzaki, “Fabrication and photoluminescence of GaInAsP/InP 2D photonic crystals,” Jpn. J. Appl. Phys., Part 2 35, 1348–1352 (1996).
[CrossRef]

Bi, W. G.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Birks, T. A.

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

Björk, G.

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. QE-27, 2386–2396 (1991).
[CrossRef]

Brand, S.

T. F. Krauss, R. M. D. L. Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[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, 3380–3383 (1991).
[CrossRef] [PubMed]

Chamberlain, K.

K. Chamberlain and L. Gordon, “Modeling good conductors using the finite-difference, time-domain technique,” IEEE Trans. Electromagn. Compat. 37, 210–216 (1995).
[CrossRef]

Chan, C. T.

C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16, 635–16, 642 (1995).
[CrossRef]

Cheng, C. C.

C. C. Cheng, A. Scherer, V. Arbet-Engels, and E. Yablonovitch, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4119 (1996).
[CrossRef]

J. O’Brien, O. Painter, C. C. Cheng, R. Lee, A. Scherer, and A. Yariv, “Lasers incorporating 2D photonic bandgap mirrors,” Electron. Lett. 32, 2243–2244 (1996).
[CrossRef]

Childs, J. J.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

Choi, D. H.

D. H. Choi and W. J. R. Hoefer, “The finite-difference-time-domain method and its application to eigenvalue problems,” IEEE Trans. Microwave Theory Tech. 34, 1464–1469 (1986).
[CrossRef]

Chu, D. Y.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
[CrossRef]

D. Y. Chu and S.-T. Ho, “Spontaneous emission from excitons in cylindrical dielectric waveguides and the spontaneous-emission factor of microcavity ring lasers,” J. Opt. Soc. Am. B 10, 381–390 (1993).
[CrossRef]

Crawford, F. D.

Y. Zou, J. S. Osinski, P. Grodzinski, P. D. Dapkus, W. Rideout, W. F. Sharfim, and F. D. Crawford, “Experimental verification of strain benefits in 1.5 μm semiconductor lasers by carrier lifetime and gain measurements,” IEEE Photonics Technol. Lett. 4, 1315–1318 (1992).
[CrossRef]

D’Urso, B.

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, 2017–2020 (1991).
[CrossRef] [PubMed]

Dapkus, P. D.

Y. Zou, J. S. Osinski, P. Grodzinski, P. D. Dapkus, W. Rideout, W. F. Sharfim, and F. D. Crawford, “Experimental verification of strain benefits in 1.5 μm semiconductor lasers by carrier lifetime and gain measurements,” IEEE Photonics Technol. Lett. 4, 1315–1318 (1992).
[CrossRef]

der Gaag, B. P. V.

B. P. V. der Gaag and A. Scherer, Appl. Phys. Lett. 56, 481–483 (1989).
[CrossRef]

Ehrfeld, W.

G. Feiertag, W. Ehrfeld, H. Freimuth, H. Kolle, H. Lehr, M. Schmidt, M. M. Sigalas, C. M. Soukoulis, G. Kiriakidis, T. Pederson, J. Kuhl, and W. Koenig, “Fabrication of photonic crystals by deep x-ray lithography,” Appl. Phys. Lett. 71, 1441–1443 (1997).
[CrossRef]

Fan, S.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Feiertag, G.

G. Feiertag, W. Ehrfeld, H. Freimuth, H. Kolle, H. Lehr, M. Schmidt, M. M. Sigalas, C. M. Soukoulis, G. Kiriakidis, T. Pederson, J. Kuhl, and W. Koenig, “Fabrication of photonic crystals by deep x-ray lithography,” Appl. Phys. Lett. 71, 1441–1443 (1997).
[CrossRef]

Feld, M. S.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

Ferrera, J.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
[CrossRef]

Florez, L. T.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, and L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. QE-27, 1332–1346 (1996).

Foresi, J. S.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
[CrossRef]

Freimuth, H.

G. Feiertag, W. Ehrfeld, H. Freimuth, H. Kolle, H. Lehr, M. Schmidt, M. M. Sigalas, C. M. Soukoulis, G. Kiriakidis, T. Pederson, J. Kuhl, and W. Koenig, “Fabrication of photonic crystals by deep x-ray lithography,” Appl. Phys. Lett. 71, 1441–1443 (1997).
[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 lattices as revealed by transmittance measurement,” Phys. Rev. B 53, 1010–1013 (1996).
[CrossRef]

Gmitter, T. J.

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J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
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A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometre radius disk laser,” Electron. Lett. 29, 1666–1667 (1993).
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D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
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J. O’Brien, O. Painter, C. C. Cheng, R. Lee, A. Scherer, and A. Yariv, “Lasers incorporating 2D photonic bandgap mirrors,” Electron. Lett. 32, 2243–2244 (1996).
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S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
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Y. Zou, J. S. Osinski, P. Grodzinski, P. D. Dapkus, W. Rideout, W. F. Sharfim, and F. D. Crawford, “Experimental verification of strain benefits in 1.5 μm semiconductor lasers by carrier lifetime and gain measurements,” IEEE Photonics Technol. Lett. 4, 1315–1318 (1992).
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G. Feiertag, W. Ehrfeld, H. Freimuth, H. Kolle, H. Lehr, M. Schmidt, M. M. Sigalas, C. M. Soukoulis, G. Kiriakidis, T. Pederson, J. Kuhl, and W. Koenig, “Fabrication of photonic crystals by deep x-ray lithography,” Appl. Phys. Lett. 71, 1441–1443 (1997).
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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
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S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
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J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
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T. Krauss, Y. P. Song, S. Thoms, C. D. W. Wilkinson, and R. M. D. L. Rue, “Fabrication of 2-D photonic bandgap structures in GaAs/AlGaAs,” Electron. Lett. 30, 1444–1446 (1994).
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T. F. Krauss, B. Vögele, C. R. Stanley, and R. M. D. L. Rue, “Waveguide microcavity based on photonic microstructures,” IEEE Photonics Technol. Lett. 9, 176–178 (1997).
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J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
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J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
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J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
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T. Krauss, Y. P. Song, S. Thoms, C. D. W. Wilkinson, and R. M. D. L. Rue, “Fabrication of 2-D photonic bandgap structures in GaAs/AlGaAs,” Electron. Lett. 30, 1444–1446 (1994).
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J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
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Tu, C. W.

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
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J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature (London) 390, 143–145 (1997).
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P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
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T. F. Krauss, B. Vögele, C. R. Stanley, and R. M. D. L. Rue, “Waveguide microcavity based on photonic microstructures,” IEEE Photonics Technol. Lett. 9, 176–178 (1997).
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K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, “Near-infrared photonic band gap of two-dimensional triangular air-rod lattices as revealed by transmittance measurement,” Phys. Rev. B 53, 1010–1013 (1996).
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T. Krauss, Y. P. Song, S. Thoms, C. D. W. Wilkinson, and R. M. D. L. Rue, “Fabrication of 2-D photonic bandgap structures in GaAs/AlGaAs,” Electron. Lett. 30, 1444–1446 (1994).
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J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, and S. T. Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photonics Technol. Lett. 8, 491–493 (1996).
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C. C. Cheng, A. Scherer, V. Arbet-Engels, and E. Yablonovitch, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4119 (1996).
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B. D’Urso, O. Painter, J. O’Brien, T. Tombrello, A. Scherer, and A. Yariv, “Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,” J. Opt. Soc. Am. B 15, 1155–1159 (1998).
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J. O’Brien, O. Painter, C. C. Cheng, R. Lee, A. Scherer, and A. Yariv, “Lasers incorporating 2D photonic bandgap mirrors,” Electron. Lett. 32, 2243–2244 (1996).
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O. Painter, J. Vučković, and A. Scherer, in an unpublished study, analyze similar microcavities, as presented in this paper, although with different bottom substrates of the waveguide and with increasing number of photonic crystal layers. Calculations of QT, Q, and Q versus frequency of the defect mode with seven layers of photonic crystal shows a smooth peak for all the Q’s, with none of the structure present for only three layers.

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Figures (18)

Fig. 1
Fig. 1

Schematic of the 2D patterned dielectric slab. The slab is surrounded by air, and the 2D array of holes completely perforate the slab.

Fig. 2
Fig. 2

(a) Top view of a microfabricated 2D hexagonal array of air holes with a single central hole missing. The interhole spacing a is 500 nm, and the radius of the holes is approximately 150 nm. (b) Cross section through the patterned membrane structure. The thickness of the undercut membrane is 180 nm, approximately one half-wavelength in the material. The membrane contains four strained quaternary quantum wells, optimized for 1.55-µm emission wavelength.

Fig. 3
Fig. 3

Schematic of a 2D slice through the middle of the patterned high-index slab. The center hole has a refractive index larger than air, nd, which forms a defect in the hexagonal lattice of air holes.

Fig. 4
Fig. 4

Band diagram for TE-polarized light (E field polarized in plane). The holes have an index nair=1. The material has an index nslab=2.65. The radius of the holes is defined by the ratio r/a=0.3. The resulting TE bandgap extends between a normalized frequency, Δωgap=a/λo=0.280.35.

Fig. 5
Fig. 5

Band diagram for TM-polarized light (E field polarized in the zˆ-direction). In this case the index contrast and r/a are not large enough to open a full 2D bandgap between the first (dielectric) and second (air) bands.

Fig. 6
Fig. 6

Band structure of the TE-like modes of the 2D patterned slab waveguide surrounded by air (d=0.4a). The solid line represents the light line. Only the guided modes are plotted.

Fig. 7
Fig. 7

Plot of the air and dielectric band edges as a function of slab thickness. The midgap frequency is also plotted, as a dashed curve.

Fig. 8
Fig. 8

Plot of the bandgap between the fundamental guided air band and dielectric band versus slab thickness.

Fig. 9
Fig. 9

Plot of the normalized frequency versus defect refractive index of the degenerate defect mode. The radius of the defect hole was kept constant at 0.3a, whereas the refractive index was varied.

Fig. 10
Fig. 10

2D slice through the middle of the slab, showing the electric-field amplitudes of the degenerate defect modes. (a) x dipole mode, (b) y dipole mode.

Fig. 11
Fig. 11

Guided in-plane radiation losses of the x and y dipole modes (degenerate case) are shown in (a) and (b), respectively. In (c) a cross section along the yˆ direction shows the radiation out the top half of the cavity for the y dipole mode. In each plot the electric-field amplitude has been enhanced to highlight the losses.

Fig. 12
Fig. 12

Plot of the quality factor versus normalized frequency of the y dipole mode.

Fig. 13
Fig. 13

Plot of the effective quality factor in the vertical direction Q versus normalized frequency.

Fig. 14
Fig. 14

Plot of the effective in-plane quality factor Q versus normalized frequency.

Fig. 15
Fig. 15

Plot of the quality factor for increasing number of photonic crystal layers that surround the defect region.

Fig. 16
Fig. 16

Cavity geometry for splitting of the dipole mode degeneracy. Only the nearest neighbor holes of the defect are shown. The two nearest holes in the x direction are enlarged. The central defect hole is filled in and has a dielectric constant equal to that of the slab. Notice that the two enlarged holes are also moved inward toward the central hole to preserve the spacing between holes in the xˆ direction.

Fig. 17
Fig. 17

(a) Plot of the calculated Q of the y dipole as a function of r/a. (b) Plot of the normalized frequency of the y dipole mode.

Fig. 18
Fig. 18

Fourier spectrum of an initial field chosen to excite both the x and y dipole modes. r/a=0.35 in this case. For larger r/a ratios the splitting of the x and y dipoles is strong enough to push the x dipole frequency out of the bandgap. The y dipole, however, is still highly localized, and its frequency is changed only moderately.

Equations (7)

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

IthqNoVactiveτsp+qγβ+Inr.
ωc2=1n2(k2+k2),
U(t)=U(0)exp(-t/τph)=U(0)exp[-(ωot)/Q],
Qωoτph.
QωoU(t)P(t).
totalpowerlosttoboundarytotalenergystoredinfield=PT(t)U(t)
=ωo1QT=P(t)+P(t)U(t)=ωo1Q+1Q.

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