Abstract

A numerical analysis of extrinsic diffraction losses in two-dimensional photonic crystal slabs with line defects is reported. To model disorder, a Gaussian distribution of hole radii in the triangular lattice of airholes is assumed. The extrinsic losses below the light line increase quadratically with the disorder parameter, decrease slightly with increasing core thickness, and depend weakly on the hole radius. For typical values of the disorder parameter the calculated loss values of guided modes below the light line compare favorably with available experimental results.

© 2004 Optical Society of America

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  1. See papers in Feature on Photonic Crystal Structures and Applications, T. F. Krauss and T. Baba, eds., IEEE J. Quantum Electron. 38, 724–963 (2002).
    [CrossRef]
  2. A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
    [CrossRef]
  3. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 62, 8212 (2000).
    [CrossRef]
  4. H.-Y. Ryu, J.-K. Hwang, and Y.-H. Lee, Phys. Rev. B 59, 5463 (1999).
    [CrossRef]
  5. Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
    [CrossRef]
  6. W. Bogaerts, P. Bienstman, and R. Baets, Opt. Lett. 28, 689 (2003).
    [CrossRef] [PubMed]
  7. L. C. Andreani and M. Agio, Appl. Phys. Lett. 82, 2011 (2003).
    [CrossRef]
  8. M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.
  9. S. J. McNab, N. Moll, and Yu. Vlasov, Opt. Express 11, 2927 (2003), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  10. L. C. Andreani and M. Agio, in Ref. 1, p. 891.
  11. L. C. Andreani, Phys. Status Solidi B 234, 139 (2002).
    [CrossRef]
  12. We typically use up to 461 plane waves and two guided modes in the basis set for air-bridge structures, taking advantage of the horizontal mirror symmetry of the slab. The number of guided modes is doubled in the case of SOI. The calculations employ a supercell in the direction ΓM perpendicular to the line defect, and an average over the results with supercell widths from 3w0+w to 8w0+w is taken to smooth out finite supercell effects, as in Ref. 7.
  13. The supercell along ΓK used to model the disorder typically has a size of 39a. Note that the use of this supercell does not require the number of plane waves to be increased, since disorder-induced scattering is treated by perturbation theory. All loss results include an average over calculations with six different random distributions corresponding to the same disorder parameter Δr/a.
  14. The defect mode shown in Fig. 1 is globally odd (σkz=-1), but its dominant field components are spatially even with respect to the vertical midplane kz bisecting the waveguide channel.
  15. K. Yamada, H. Morita, A. Shinya, and M. Notomi, Opt. Commun. 198, 395 (2001).
    [CrossRef]
  16. M. Qiu, Phys. Rev. B 66, 033103 (2002).
    [CrossRef]
  17. S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
    [CrossRef]

2003 (4)

Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
[CrossRef]

W. Bogaerts, P. Bienstman, and R. Baets, Opt. Lett. 28, 689 (2003).
[CrossRef] [PubMed]

L. C. Andreani and M. Agio, Appl. Phys. Lett. 82, 2011 (2003).
[CrossRef]

S. J. McNab, N. Moll, and Yu. Vlasov, Opt. Express 11, 2927 (2003), http://www.opticsexpress.org .
[CrossRef] [PubMed]

2002 (3)

L. C. Andreani, Phys. Status Solidi B 234, 139 (2002).
[CrossRef]

See papers in Feature on Photonic Crystal Structures and Applications, T. F. Krauss and T. Baba, eds., IEEE J. Quantum Electron. 38, 724–963 (2002).
[CrossRef]

M. Qiu, Phys. Rev. B 66, 033103 (2002).
[CrossRef]

2001 (1)

K. Yamada, H. Morita, A. Shinya, and M. Notomi, Opt. Commun. 198, 395 (2001).
[CrossRef]

2000 (2)

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 62, 8212 (2000).
[CrossRef]

1999 (1)

H.-Y. Ryu, J.-K. Hwang, and Y.-H. Lee, Phys. Rev. B 59, 5463 (1999).
[CrossRef]

Agio, M.

L. C. Andreani and M. Agio, Appl. Phys. Lett. 82, 2011 (2003).
[CrossRef]

L. C. Andreani and M. Agio, in Ref. 1, p. 891.

Akahane, Y.

Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
[CrossRef]

Andreani, L. C.

L. C. Andreani and M. Agio, Appl. Phys. Lett. 82, 2011 (2003).
[CrossRef]

L. C. Andreani, Phys. Status Solidi B 234, 139 (2002).
[CrossRef]

L. C. Andreani and M. Agio, in Ref. 1, p. 891.

Asano, T.

Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
[CrossRef]

Baets, R.

Bienstman, P.

W. Bogaerts, P. Bienstman, and R. Baets, Opt. Lett. 28, 689 (2003).
[CrossRef] [PubMed]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

Bogaerts, W.

Chutinan, A.

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 62, 8212 (2000).
[CrossRef]

Hwang, J.-K.

H.-Y. Ryu, J.-K. Hwang, and Y.-H. Lee, Phys. Rev. B 59, 5463 (1999).
[CrossRef]

Ibanescu, M.

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 62, 8212 (2000).
[CrossRef]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

Johnson, S. G.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 62, 8212 (2000).
[CrossRef]

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

Lee, Y.-H.

H.-Y. Ryu, J.-K. Hwang, and Y.-H. Lee, Phys. Rev. B 59, 5463 (1999).
[CrossRef]

Lidorikis, E.

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

McNab, S. J.

Moll, N.

Morita, H.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, Opt. Commun. 198, 395 (2001).
[CrossRef]

Noda, S.

Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
[CrossRef]

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Notomi, M.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, Opt. Commun. 198, 395 (2001).
[CrossRef]

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.

Povinelli, M. L.

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

Qiu, M.

M. Qiu, Phys. Rev. B 66, 033103 (2002).
[CrossRef]

Ryu, H.-Y.

H.-Y. Ryu, J.-K. Hwang, and Y.-H. Lee, Phys. Rev. B 59, 5463 (1999).
[CrossRef]

Shinya, A.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, Opt. Commun. 198, 395 (2001).
[CrossRef]

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.

Skorobogatiy, M.

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

Soljacic, M.

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

Song, B.-S.

Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
[CrossRef]

Takahashi, C.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.

Takahashi, J.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.

Tanaka, Y.

Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 62, 8212 (2000).
[CrossRef]

Vlasov, Yu.

Yamada, K.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, Opt. Commun. 198, 395 (2001).
[CrossRef]

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.

Yokohama, I.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.

Appl. Phys. Lett. (2)

Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, Appl. Phys. Lett. 82, 1661 (2003).
[CrossRef]

L. C. Andreani and M. Agio, Appl. Phys. Lett. 82, 2011 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

See papers in Feature on Photonic Crystal Structures and Applications, T. F. Krauss and T. Baba, eds., IEEE J. Quantum Electron. 38, 724–963 (2002).
[CrossRef]

Opt. Commun. (1)

K. Yamada, H. Morita, A. Shinya, and M. Notomi, Opt. Commun. 198, 395 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (4)

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 62, 8212 (2000).
[CrossRef]

H.-Y. Ryu, J.-K. Hwang, and Y.-H. Lee, Phys. Rev. B 59, 5463 (1999).
[CrossRef]

M. Qiu, Phys. Rev. B 66, 033103 (2002).
[CrossRef]

Phys. Status Solidi B (1)

L. C. Andreani, Phys. Status Solidi B 234, 139 (2002).
[CrossRef]

Other (6)

We typically use up to 461 plane waves and two guided modes in the basis set for air-bridge structures, taking advantage of the horizontal mirror symmetry of the slab. The number of guided modes is doubled in the case of SOI. The calculations employ a supercell in the direction ΓM perpendicular to the line defect, and an average over the results with supercell widths from 3w0+w to 8w0+w is taken to smooth out finite supercell effects, as in Ref. 7.

The supercell along ΓK used to model the disorder typically has a size of 39a. Note that the use of this supercell does not require the number of plane waves to be increased, since disorder-induced scattering is treated by perturbation theory. All loss results include an average over calculations with six different random distributions corresponding to the same disorder parameter Δr/a.

The defect mode shown in Fig. 1 is globally odd (σkz=-1), but its dominant field components are spatially even with respect to the vertical midplane kz bisecting the waveguide channel.

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, in Proceedings of 2003 Fifth International Conference on Transparent Optical Networks (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), p. 103.
[CrossRef]

L. C. Andreani and M. Agio, in Ref. 1, p. 891.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, in Ref. 1, p. 736.

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Figures (4)

Fig. 1
Fig. 1

(a) Dispersion of the defect mode; the dotted line is the air light line. (b) Imaginary part of the frequency for different values of disorder parameter Δr/a. Parameters of the W1 air-bridge structure are r/a=0.28, d/a=0.5, core=12, air=1.

Fig. 2
Fig. 2

(a) Imaginary part of the frequency for different core thicknesses of a W1 air bridge with r/a=0.28. (b) Imaginary part of the frequency for different hole radii of a W1 air bridge with d/a=0.5. The disorder parameter is set equal to the typical value of Δr/a=0.01.

Fig. 3
Fig. 3

Comparison between W1 (solid curves and filled circles) and W07 (dashed curves and open circles) linear waveguides in membrane and SOI structures, respectively. (a) Dispersion of the defect mode, group velocity, imaginary part of the frequency, and losses (dimensionless losses, αa in decibels) for an air-bridge PhC waveguide with r/a=0.28, d/a=0.5, Δr/a=0.01; (b) same quantities for a SOI-based PhC waveguide with identical structure parameters and ox=2.1 for the SiO2 substrate. The results for the group velocity, imaginary part of the frequency, and losses are plot-ted in only the energy range for which the defect mode lies below the light line (or below the parity mixing region for SOI).

Fig. 4
Fig. 4

(a) Defect mode dispersion and (b) propagation losses for the silicon membrane structure studied in Ref. 9: W1 waveguide, a=445 nm, r/a=0.37, d/a=0.5, nominal disorder parameter Δr=5 nm, Si=12.

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