Abstract

Waveguide modes in two-dimensional (2-D) photonic crystals (PhCs) deeply etched through monomode slab waveguides, e.g., AlGaAs/GaAs, GaAs/AlOx, or InP/GaInAsP, suffer from radiation losses that are strongly affected by the air hole depth and shape. The issue of three-dimensional (3-D) out-of-plane losses is addressed analytically by means of an incoherent approximation. Assuming separability both for the dielectric map and for the electric field, this approach is valid for defects such as in-plane microcavities, PhC-based waveguides, bends and couplers. Out-of-plane scattering is translated into an effective imaginary index in the air holes, so that 3-D losses can be cast in a simple 2-D calculation. The case of cylindroconical holes is treated, and the validity of this approach is experimentally confirmed by transmission measurements through simple PhC slabs.

© 2003 Optical Society of America

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  3. A. Sharkawy, S. Shi, and D. Prather, “Multichannel wavelength division multiplexing with photonic crystals,” Appl. Opt. 40, 2247–2252 (2001).
    [CrossRef]
  4. U. Gruning, 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]
  5. S. W. Leonard, H. M. van Driel, K. Busch, S. John, A. Birner, A. P. Li, F. Muller, U. Gosele, and V. Lehmann, “Attenuation of optical transmission within the band gap of thin two-dimensional macroporous silicon photonic crystals,” Appl. Phys. Lett. 75, 3063–3065 (1999).
    [CrossRef]
  6. S. Rowson, A. Chelnokov, and J. M. Lourtioz, “Two-dimensional photonic crystals in macroporous silicon: from mid-infrared (10 μm) to telecommunication wavelengths (1.3–1.5 μm),” J. Lightwave Technol. 17, 1989–1995 (1999).
    [CrossRef]
  7. R. B. Wehrspohn, A. Birner, J. Schilling, F. Mueller, R. Hillebrand, and U. Gösele, “Photonic crystals from macroporous silicon,” in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 143–153.
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    [CrossRef]
  9. A. Scherer, O. Painter, B. D’Urso, R. Lee, and A. Yariv, “InGaAsP photonic band gap crystal membrane microresonators,” J. Vac. Sci. Technol. B 16, 3906–3910 (1998).
    [CrossRef]
  10. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
    [CrossRef]
  11. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
    [CrossRef]
  12. H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. De la Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
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  15. P. Lalanne and H. Benisty, “Out-of-plane losses of two-dimensional photonic crystals waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
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  17. W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photonics Technol. Lett. 13, 565–567 (2001).
    [CrossRef]
  18. S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
    [CrossRef]
  19. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
    [CrossRef] [PubMed]
  20. N. Fukaya, D. Ohsaki, and T. Baba, “Two-dimensional photonic crystal waveguides with 60 degrees bends in a thin slab structure,” Jpn. J. Appl. Phys. 39, 2619–2623 (2000).
    [CrossRef]
  21. A. Shinya, M. Notomi, I. Yokohama, C. Takahashi, J. I. Takahashi, and T. Tamamura, “Two-dimensional Si photonic crystals on oxide using SOI substrate,” Opt. Quantum Electron. 34, 113–121 (2002).
    [CrossRef]
  22. X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. 79 (15), 2312–2314 (2001).
    [CrossRef]
  23. N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
    [CrossRef]
  24. T. F. Krauss, R. M. DeLaRue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near infrared wavelengths,” Nature 383, 699–702 (1996).
    [CrossRef]
  25. S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
    [CrossRef]
  26. A. Talneau, L. Le Gouezigou, and N. Bouadma, “Quantitative measurement of low propagation losses at 1.55 μm on planar photonic crystal waveguides,” Opt. Lett. 26, 1259–1261 (2001).
    [CrossRef]
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  28. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, and C. Jouanin, “Quantitative measurement of transmission, reflection, and diffraction of two-dimensional photonic band gap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
    [CrossRef]
  29. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
    [CrossRef]
  30. B. D’Urso, O. Painter, J. 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]
  31. M. Palamaru and P. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466–1468 (2001).
    [CrossRef]
  32. D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Robert, “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]
  33. The slope of the curve in Fig. 7 of Ref. 16.
  34. M. Qiu, B. Jaskorzynska, M. Swillo, and H. Benisty, “Time-domain 2D modeling of slab-waveguide based photonic-crystal devices in the presence of out-of-plane radiation losses,” Microwave Opt. Technol. Lett. 34, 387–393 (2002).
    [CrossRef]
  35. Given a refractive-index contrast of nguide/nair =3, the average optical path of the guided wave inside the air holes is less than 10% for f≤0.30.
  36. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  37. H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996).
    [CrossRef] [PubMed]
  38. L. C. Andreani and M. Agio, “Photonic bands and gap maps in a photonic crystal slab,” IEEE J. Quantum Electron. 38, 891–898 (2002).
    [CrossRef]
  39. D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. H. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, and L. A. Kolodziejski, “One-dimensional photonic bandgap microcavities for strong optical confinement in GaAs and GaAs/AlxOy semiconductor waveguides,” J. Lightwave Technol. 17, 2152–2160 (1999).
    [CrossRef]
  40. J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
    [CrossRef]
  41. M. Qiu and S. L. He, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B 61, 12871–12876 (2000).
    [CrossRef]
  42. M. Qiu and S. L. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
    [CrossRef]
  43. M. Qiu and S. L. He, “Guided modes in a two-dimensional metallic photonic crystal waveguide,” Phys. Lett. A 266, 425–429 (2000).
    [CrossRef]
  44. M. Qiu and S. L. He, “FDTD algorithm for computing the off-plane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions,” Phys. Lett. A 278, 348–354 (2001).
    [CrossRef]
  45. The effective air fill factor value obtained from the fit is checked to fall within the range set by the SEM analysis that is influenced by the local hole shape fluctuations that are due to sample preparation, by the average on a limited number of holes, and by the cylindroconical hole shape.

2002 (7)

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, “Optical study of two-dimensional InP-based photonic crystals by internal light source technique,” IEEE J. Quantum Electron. 38, 786–799 (2002).
[CrossRef]

H. Benisty, P. Lalanne, S. Olivier, M. Rattier, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, “Finite-depth and intrinsic losses in vertically etched two-dimensional photonic crystals,” Opt. Quantum Electron. 34, 205–215 (2002).
[CrossRef]

A. Shinya, M. Notomi, I. Yokohama, C. Takahashi, J. I. Takahashi, and T. Tamamura, “Two-dimensional Si photonic crystals on oxide using SOI substrate,” Opt. Quantum Electron. 34, 113–121 (2002).
[CrossRef]

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
[CrossRef]

M. Qiu, B. Jaskorzynska, M. Swillo, and H. Benisty, “Time-domain 2D modeling of slab-waveguide based photonic-crystal devices in the presence of out-of-plane radiation losses,” Microwave Opt. Technol. Lett. 34, 387–393 (2002).
[CrossRef]

L. C. Andreani and M. Agio, “Photonic bands and gap maps in a photonic crystal slab,” IEEE J. Quantum Electron. 38, 891–898 (2002).
[CrossRef]

J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
[CrossRef]

2001 (9)

A. Sharkawy, S. Shi, and D. Prather, “Multichannel wavelength division multiplexing with photonic crystals,” Appl. Opt. 40, 2247–2252 (2001).
[CrossRef]

A. Talneau, L. Le Gouezigou, and N. Bouadma, “Quantitative measurement of low propagation losses at 1.55 μm on planar photonic crystal waveguides,” Opt. Lett. 26, 1259–1261 (2001).
[CrossRef]

P. Lalanne and H. Benisty, “Out-of-plane losses of two-dimensional photonic crystals waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
[CrossRef]

M. Palamaru and P. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466–1468 (2001).
[CrossRef]

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
[CrossRef]

J. Schilling, R. B. Wehrspohn, A. Birner, F. Muller, R. Hillebrand, U. Gosele, S. W. Leonard, J. P. Mondia, F. Genereux, H. M. van Driel, P. Kramper, V. Sandoghdar, and K. Busch, “A model system for two-dimensional and three-dimensional photonic crystals: macroporous silicon,” J. Opt. A. Pure Appl. Opt. 3, S121–S132 (2001).
[CrossRef]

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. 79 (15), 2312–2314 (2001).
[CrossRef]

W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photonics Technol. Lett. 13, 565–567 (2001).
[CrossRef]

M. Qiu and S. L. He, “FDTD algorithm for computing the off-plane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions,” Phys. Lett. A 278, 348–354 (2001).
[CrossRef]

2000 (7)

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

N. Fukaya, D. Ohsaki, and T. Baba, “Two-dimensional photonic crystal waveguides with 60 degrees bends in a thin slab structure,” Jpn. J. Appl. Phys. 39, 2619–2623 (2000).
[CrossRef]

M. Qiu and S. L. He, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B 61, 12871–12876 (2000).
[CrossRef]

M. Qiu and S. L. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[CrossRef]

M. Qiu and S. L. He, “Guided modes in a two-dimensional metallic photonic crystal waveguide,” Phys. Lett. A 266, 425–429 (2000).
[CrossRef]

1999 (6)

S. Rowson, A. Chelnokov, and J. M. Lourtioz, “Two-dimensional photonic crystals in macroporous silicon: from mid-infrared (10 μm) to telecommunication wavelengths (1.3–1.5 μm),” J. Lightwave Technol. 17, 1989–1995 (1999).
[CrossRef]

H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. De la Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
[CrossRef]

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. H. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, and L. A. Kolodziejski, “One-dimensional photonic bandgap microcavities for strong optical confinement in GaAs and GaAs/AlxOy semiconductor waveguides,” J. Lightwave Technol. 17, 2152–2160 (1999).
[CrossRef]

S. W. Leonard, H. M. van Driel, K. Busch, S. John, A. Birner, A. P. Li, F. Muller, U. Gosele, and V. Lehmann, “Attenuation of optical transmission within the band gap of thin two-dimensional macroporous silicon photonic crystals,” Appl. Phys. Lett. 75, 3063–3065 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

1998 (2)

A. Scherer, O. Painter, B. D’Urso, R. Lee, and A. Yariv, “InGaAsP photonic band gap crystal membrane microresonators,” J. Vac. Sci. Technol. B 16, 3906–3910 (1998).
[CrossRef]

B. D’Urso, O. Painter, J. 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]

1997 (2)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, and C. Jouanin, “Quantitative measurement of transmission, reflection, and diffraction of two-dimensional photonic band gap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

1996 (4)

U. Gruning, 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]

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

H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996).
[CrossRef] [PubMed]

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Robert, “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]

1991 (1)

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

Agio, M.

L. C. Andreani and M. Agio, “Photonic bands and gap maps in a photonic crystal slab,” IEEE J. Quantum Electron. 38, 891–898 (2002).
[CrossRef]

Alleman, A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Anand, S.

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, “Optical study of two-dimensional InP-based photonic crystals by internal light source technique,” IEEE J. Quantum Electron. 38, 786–799 (2002).
[CrossRef]

J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
[CrossRef]

Andreani, L. C.

L. C. Andreani and M. Agio, “Photonic bands and gap maps in a photonic crystal slab,” IEEE J. Quantum Electron. 38, 891–898 (2002).
[CrossRef]

Asakawa, K.

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
[CrossRef]

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
[CrossRef]

Atkin, D. M.

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Robert, “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.

N. Fukaya, D. Ohsaki, and T. Baba, “Two-dimensional photonic crystal waveguides with 60 degrees bends in a thin slab structure,” Jpn. J. Appl. Phys. 39, 2619–2623 (2000).
[CrossRef]

Baets, R.

W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photonics Technol. Lett. 13, 565–567 (2001).
[CrossRef]

Bardinal, V.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, and C. Jouanin, “Quantitative measurement of transmission, reflection, and diffraction of two-dimensional photonic band gap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Benisty, H.

H. Benisty, P. Lalanne, S. Olivier, M. Rattier, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, “Finite-depth and intrinsic losses in vertically etched two-dimensional photonic crystals,” Opt. Quantum Electron. 34, 205–215 (2002).
[CrossRef]

M. Qiu, B. Jaskorzynska, M. Swillo, and H. Benisty, “Time-domain 2D modeling of slab-waveguide based photonic-crystal devices in the presence of out-of-plane radiation losses,” Microwave Opt. Technol. Lett. 34, 387–393 (2002).
[CrossRef]

P. Lalanne and H. Benisty, “Out-of-plane losses of two-dimensional photonic crystals waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. De la Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, and C. Jouanin, “Quantitative measurement of transmission, reflection, and diffraction of two-dimensional photonic band gap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Beraud, A.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

Berggren, J.

J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
[CrossRef]

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W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photonics Technol. Lett. 13, 565–567 (2001).
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D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Robert, “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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Birner, A.

J. Schilling, R. B. Wehrspohn, A. Birner, F. Muller, R. Hillebrand, U. Gosele, S. W. Leonard, J. P. Mondia, F. Genereux, H. M. van Driel, P. Kramper, V. Sandoghdar, and K. Busch, “A model system for two-dimensional and three-dimensional photonic crystals: macroporous silicon,” J. Opt. A. Pure Appl. Opt. 3, S121–S132 (2001).
[CrossRef]

S. W. Leonard, H. M. van Driel, K. Busch, S. John, A. Birner, A. P. Li, F. Muller, U. Gosele, and V. Lehmann, “Attenuation of optical transmission within the band gap of thin two-dimensional macroporous silicon photonic crystals,” Appl. Phys. Lett. 75, 3063–3065 (1999).
[CrossRef]

Bogaerts, W.

W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photonics Technol. Lett. 13, 565–567 (2001).
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Bouadma, N.

Brand, S.

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

Busch, K.

J. Schilling, R. B. Wehrspohn, A. Birner, F. Muller, R. Hillebrand, U. Gosele, S. W. Leonard, J. P. Mondia, F. Genereux, H. M. van Driel, P. Kramper, V. Sandoghdar, and K. Busch, “A model system for two-dimensional and three-dimensional photonic crystals: macroporous silicon,” J. Opt. A. Pure Appl. Opt. 3, S121–S132 (2001).
[CrossRef]

S. W. Leonard, H. M. van Driel, K. Busch, S. John, A. Birner, A. P. Li, F. Muller, U. Gosele, and V. Lehmann, “Attenuation of optical transmission within the band gap of thin two-dimensional macroporous silicon photonic crystals,” Appl. Phys. Lett. 75, 3063–3065 (1999).
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U. Gruning, 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).
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N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
[CrossRef]

Cassagne, D.

H. Benisty, P. Lalanne, S. Olivier, M. Rattier, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, “Finite-depth and intrinsic losses in vertically etched two-dimensional photonic crystals,” Opt. Quantum Electron. 34, 205–215 (2002).
[CrossRef]

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. 79 (15), 2312–2314 (2001).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. De la Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, and C. Jouanin, “Quantitative measurement of transmission, reflection, and diffraction of two-dimensional photonic band gap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
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Chelnokov, A.

Chow, E.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
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B. D’Urso, O. Painter, J. 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).
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A. Scherer, O. Painter, B. D’Urso, R. Lee, and A. Yariv, “InGaAsP photonic band gap crystal membrane microresonators,” J. Vac. Sci. Technol. B 16, 3906–3910 (1998).
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X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. 79 (15), 2312–2314 (2001).
[CrossRef]

De la Rue, R. M.

De Zutter, D.

W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photonics Technol. Lett. 13, 565–567 (2001).
[CrossRef]

DeLaRue, R. M.

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

Fan, S. H.

Ferrini, R.

J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, “Optical study of two-dimensional InP-based photonic crystals by internal light source technique,” IEEE J. Quantum Electron. 38, 786–799 (2002).
[CrossRef]

Forchel, A.

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, “Optical study of two-dimensional InP-based photonic crystals by internal light source technique,” IEEE J. Quantum Electron. 38, 786–799 (2002).
[CrossRef]

J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
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Fukaya, N.

N. Fukaya, D. Ohsaki, and T. Baba, “Two-dimensional photonic crystal waveguides with 60 degrees bends in a thin slab structure,” Jpn. J. Appl. Phys. 39, 2619–2623 (2000).
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J. Schilling, R. B. Wehrspohn, A. Birner, F. Muller, R. Hillebrand, U. Gosele, S. W. Leonard, J. P. Mondia, F. Genereux, H. M. van Driel, P. Kramper, V. Sandoghdar, and K. Busch, “A model system for two-dimensional and three-dimensional photonic crystals: macroporous silicon,” J. Opt. A. Pure Appl. Opt. 3, S121–S132 (2001).
[CrossRef]

Gosele, U.

J. Schilling, R. B. Wehrspohn, A. Birner, F. Muller, R. Hillebrand, U. Gosele, S. W. Leonard, J. P. Mondia, F. Genereux, H. M. van Driel, P. Kramper, V. Sandoghdar, and K. Busch, “A model system for two-dimensional and three-dimensional photonic crystals: macroporous silicon,” J. Opt. A. Pure Appl. Opt. 3, S121–S132 (2001).
[CrossRef]

S. W. Leonard, H. M. van Driel, K. Busch, S. John, A. Birner, A. P. Li, F. Muller, U. Gosele, and V. Lehmann, “Attenuation of optical transmission within the band gap of thin two-dimensional macroporous silicon photonic crystals,” Appl. Phys. Lett. 75, 3063–3065 (1999).
[CrossRef]

Grillet, C.

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. 79 (15), 2312–2314 (2001).
[CrossRef]

Gruning, U.

U. Gruning, 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]

He, S. L.

M. Qiu and S. L. He, “FDTD algorithm for computing the off-plane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions,” Phys. Lett. A 278, 348–354 (2001).
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M. Qiu and S. L. He, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B 61, 12871–12876 (2000).
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M. Qiu and S. L. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
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M. Qiu and S. L. He, “Guided modes in a two-dimensional metallic photonic crystal waveguide,” Phys. Lett. A 266, 425–429 (2000).
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J. Schilling, R. B. Wehrspohn, A. Birner, F. Muller, R. Hillebrand, U. Gosele, S. W. Leonard, J. P. Mondia, F. Genereux, H. M. van Driel, P. Kramper, V. Sandoghdar, and K. Busch, “A model system for two-dimensional and three-dimensional photonic crystals: macroporous silicon,” J. Opt. A. Pure Appl. Opt. 3, S121–S132 (2001).
[CrossRef]

Hou, H.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

Houdré, R.

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, “Optical study of two-dimensional InP-based photonic crystals by internal light source technique,” IEEE J. Quantum Electron. 38, 786–799 (2002).
[CrossRef]

J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
[CrossRef]

H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. De la Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, and C. Jouanin, “Quantitative measurement of transmission, reflection, and diffraction of two-dimensional photonic band gap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Ikeda, N.

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
[CrossRef]

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
[CrossRef]

Inoue, K.

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
[CrossRef]

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
[CrossRef]

Ippen, E. P.

Jaskorzynska, B.

M. Qiu, B. Jaskorzynska, M. Swillo, and H. Benisty, “Time-domain 2D modeling of slab-waveguide based photonic-crystal devices in the presence of out-of-plane radiation losses,” Microwave Opt. Technol. Lett. 34, 387–393 (2002).
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Joannopoulos, J. D.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
[CrossRef]

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. H. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, and L. A. Kolodziejski, “One-dimensional photonic bandgap microcavities for strong optical confinement in GaAs and GaAs/AlxOy semiconductor waveguides,” J. Lightwave Technol. 17, 2152–2160 (1999).
[CrossRef]

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

John, S.

S. W. Leonard, H. M. van Driel, K. Busch, S. John, A. Birner, A. P. Li, F. Muller, U. Gosele, and V. Lehmann, “Attenuation of optical transmission within the band gap of thin two-dimensional macroporous silicon photonic crystals,” Appl. Phys. Lett. 75, 3063–3065 (1999).
[CrossRef]

Johnson, S. G.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[CrossRef] [PubMed]

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
[CrossRef]

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Jouanin, C.

H. Benisty, P. Lalanne, S. Olivier, M. Rattier, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, “Finite-depth and intrinsic losses in vertically etched two-dimensional photonic crystals,” Opt. Quantum Electron. 34, 205–215 (2002).
[CrossRef]

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. 79 (15), 2312–2314 (2001).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. De la Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, and C. Jouanin, “Quantitative measurement of transmission, reflection, and diffraction of two-dimensional photonic band gap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Kamp, M.

J. Moosburger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Berggren, “Nanofabrication of high quality photonic crystals for integrated optics circuits,” Nanotechnology 13, 341–345 (2002).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, “Optical study of two-dimensional InP-based photonic crystals by internal light source technique,” IEEE J. Quantum Electron. 38, 786–799 (2002).
[CrossRef]

Katayama, Y.

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
[CrossRef]

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
[CrossRef]

Kawai, N.

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, “Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures,” Opt. Quantum Electron. 34, 123–131 (2002).
[CrossRef]

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
[CrossRef]

Kolodziejski, L. A.

Koyama, T.

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001).
[CrossRef]

Kramper, P.

J. Schilling, R. B. Wehrspohn, A. Birner, F. Muller, R. Hillebrand, U. Gosele, S. W. Leonard, J. P. Mondia, F. Genereux, H. M. van Driel, P. Kramper, V. Sandoghdar, and K. Busch, “A model system for two-dimensional and three-dimensional photonic crystals: macroporous silicon,” J. Opt. A. Pure Appl. Opt. 3, S121–S132 (2001).
[CrossRef]

Krauss, T. F.

H. Benisty, P. Lalanne, S. Olivier, M. Rattier, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, “Finite-depth and intrinsic losses in vertically etched two-dimensional photonic crystals,” Opt. Quantum Electron. 34, 205–215 (2002).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdré, U. Oesterle, and V. Bardinal, “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. De la Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
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R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosburger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, “Optical study of two-dimensional InP-based photonic crystals by internal light source technique,” IEEE J. Quantum Electron. 38, 786–799 (2002).
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Other (7)

R. B. Wehrspohn, A. Birner, J. Schilling, F. Mueller, R. Hillebrand, and U. Gösele, “Photonic crystals from macroporous silicon,” in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 143–153.

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The effective air fill factor value obtained from the fit is checked to fall within the range set by the SEM analysis that is influenced by the local hole shape fluctuations that are due to sample preparation, by the average on a limited number of holes, and by the cylindroconical hole shape.

Given a refractive-index contrast of nguide/nair =3, the average optical path of the guided wave inside the air holes is less than 10% for f≤0.30.

The slope of the curve in Fig. 7 of Ref. 16.

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Figures (7)

Fig. 1
Fig. 1

Sketch of an ideal 2-D PhC etched through a conventional semiconductor step-index waveguide with a symmetric dielectric constant profile ε1/ε2/ε1. The PhC consists of a triangular array of infinitely deep air holes. The top cladding thickness z1, the core width (z2z1), and the guided-mode profile ζ(z) are also shown.

Fig. 2
Fig. 2

(a) Same as Fig. 1 for the real case of a 2-D PhC with a finite hole depth z0. The partial confinement factor Γ(z0) is also shown, i.e., the overlap integral of the squared field profile ζ2(z) with the missing air column region [see Eq. (14)]; (b) sketch of the dielectric perturbation with respect to the case of infinitely deep holes.

Fig. 3
Fig. 3

Estimate of the loss parameter εhole=BΓ(z0) as a function of the etched depth |z0| for hole diameters that correspond to typical air-fill factor values of f=0.200.30. Three basic material systems are considered: (a) a typical GaAs/AlOx-based 2-D PhC, (b) an AlxGa1xAs/GaAs structure, (c) an InP/GaInAsP system. The curves are shown only for z0 values lying in the bottom cladding (i.e., |z0|>|z2|; see Figs. 1 and 2). The partial confinement factor Γ(z0) is plotted in the inset for the three cases.

Fig. 4
Fig. 4

Sketch of the cylindroconical hole shape model. The main geometric parameters and the guided field profile ζ (z) are shown. zeq is defined as the equivalent depth of a cylindrical hole that gives the same amount of losses as the tapered-bottom hole.

Fig. 5
Fig. 5

Illustration of three different cases for losses with cylindroconical holes: (a) cone height Δz much smaller than the decay length Ldecay1/K of the guided mode: flat bottom limit; (b) intermediate case: ΔzLdecay; (c) ΔzLdecay: vertical wall limit; (d) plot of the normalized equivalent depth K(zb-zeq) as a function of the normalized total depth KΔz.

Fig. 6
Fig. 6

(a) Sketch of the InP/GaInAsP waveguide (WG) heterostructure. The cylindroconical hole shape deduced from the SEM analysis of the sample is illustrated. (b) Edge view SEM micrograph of a test PhC structure for the studied sample. The image was taken before the removal of the SiO2 mask. In the inset the almost cylindrical shape of the holes down to z2 is evident.

Fig. 7
Fig. 7

(a) TE transmission spectra through ΓM and ΓK oriented PhC slabs for the sample in Fig. 6. Experimental spectra (boldface curves) are compared with 2-D FDTD calculated spectra (lightface curves). (b) Sketch of the typical layout of simple PhC structures along ΓM and ΓK orientations. Each slab is eight rows thick and is characterized by the period a (or hole diameter D) value. Hairlines show a single atomic plane for both orientations.

Equations (25)

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ε=εint+εhole,
E(x, y, z)=ψ(x, y)ζ(z).
P(x, y, z)=E(x, y, z)ε0ε˜=ψ(x, y)ζ(z)ε0ε˜,
P(x, y, z)exp(+Kz)
p=plugvolumeP(x, y, z)dxdydz.
Prad=p2ω4n112πε0c3
Prad
=12ε0ω46πc3 n1ε˜2Sψ(x, y)dxdy2[-z0ζ(z)dz]2.
Pdiss=12 ε0ωεholeholesE2(x, y, z)dxdydz=12 ε0ωεholeSψ2(x, y)dxdy-+ζ2(z)dz,
Sψ(x, y)dxdy2Shole×Sψ2(x, y)dxdy.
εhole=ε˜216πc3 ω3n1-z0ζ(z)dz2-+ζ2(z)dz Shole.
ζ(z)=A exp(+Kz),
-z0ζ(z)dz2=2K-z0ζ2(z)dz.
εhole=ε˜28π23λ3 n1Γ(z0)LdecayShole,
Γ(z0)=-z0ζ2(z)dz-+ζ2(z)dz
-z0ζ(z)dz-zbg(z)ζ(z)dz,
g(z)=1(|z|>|za|),
g(z)=1-r(z)2r2(|zb|<|z|<|za|),
-zb g(z)ζ(z)dz=-zaexp[-K(z-za)]dz+zazb1-r(z)2r2×exp[-K(z-za)]dz.
-zb g(z)ζ(z)dz=2Kexp(KΔz)1KΔz-1-exp(-KΔz)K2Δz2.
-zeqζ(z)dz=1Kexp[K(zeq-zb)]=1Kexp(KΔz)exp[K(zeq-zb)].
exp[-K(zb-zeq)]=21KΔz-1-exp(-KΔz)K2Δz2.
zb-zeq=Δz3,
zb-zeq=-1Kln2KΔz,
εintwλ/n2 (u2f )(Δε)2ηΓ2,

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