Abstract

The phenomenological approach introduced by Benisty et al. [Appl. Phys. Lett. 76, 532 (2000) ] to model out-of-plane radiation losses in planar photonic crystals with a low vertical refractive index contrast is extended to the case of in-plane disorder. The model is experimentally validated by means of optical measurements on GaAs-based structures. For the present fabrication techniques the disorder-induced contribution is found to be negligible compared with the other loss mechanisms.

© 2006 Optical Society of America

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  1. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
    [CrossRef]
  2. W. Bogaerts, P. Bienstman, and R. Baets, Opt. Lett. 28, 689 (2003).
    [CrossRef] [PubMed]
  3. M. Skorobogatiy, G. Bégin, and A. Talneau, Opt. Express 13, 2487 (2005), and references therein.
    [CrossRef] [PubMed]
  4. J. M. Rico-García, J. M. López-Alonso, and J. Alda, Opt. Express 13, 3082 (2005), and references therein.
    [CrossRef]
  5. H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
    [CrossRef]
  6. R. Ferrini, R. Houdré, H. Benisty, M. Qiu, and J. Moosbürger, J. Opt. Soc. Am. B 20, 469 (2003).
    [CrossRef]
  7. R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
    [CrossRef]
  8. J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
    [CrossRef]
  9. R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
    [CrossRef]
  10. W. Whelan-Curtin, X. Yuan, and T. F. Krauss, 'Effects of disorder in photonic crystals,' presented at Proceedings of the PECS-VI Conference, Crete, 2005.
  11. D. Gerace and L. C. Andreani, Photonics Nanostruct. Fundam. Appl. 3, 120 (2005).
    [CrossRef]
  12. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005).
    [CrossRef] [PubMed]
  13. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
    [CrossRef]

2005

J. M. Rico-García, J. M. López-Alonso, and J. Alda, Opt. Express 13, 3082 (2005), and references therein.
[CrossRef]

D. Gerace and L. C. Andreani, Photonics Nanostruct. Fundam. Appl. 3, 120 (2005).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

M. Skorobogatiy, G. Bégin, and A. Talneau, Opt. Express 13, 2487 (2005), and references therein.
[CrossRef] [PubMed]

2004

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

2003

2002

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

2000

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
[CrossRef]

Alda, J.

J. M. Rico-García, J. M. López-Alonso, and J. Alda, Opt. Express 13, 3082 (2005), and references therein.
[CrossRef]

Anand, S.

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Andreani, L. C.

D. Gerace and L. C. Andreani, Photonics Nanostruct. Fundam. Appl. 3, 120 (2005).
[CrossRef]

Baets, R.

Bégin, G.

Benisty, H.

R. Ferrini, R. Houdré, H. Benisty, M. Qiu, and J. Moosbürger, J. Opt. Soc. Am. B 20, 469 (2003).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Bergreen, J.

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Berrier, A.

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

Bienstman, P.

Bogaerts, W.

Cassagne, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

de Rossi, S.

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

Dunbar, L. A.

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

Economou, E. N.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
[CrossRef]

Ferrini, R.

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

R. Ferrini, R. Houdré, H. Benisty, M. Qiu, and J. Moosbürger, J. Opt. Soc. Am. B 20, 469 (2003).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Forchel, A.

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

Gerace, D.

D. Gerace and L. C. Andreani, Photonics Nanostruct. Fundam. Appl. 3, 120 (2005).
[CrossRef]

Houdré, R.

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

R. Ferrini, R. Houdré, H. Benisty, M. Qiu, and J. Moosbürger, J. Opt. Soc. Am. B 20, 469 (2003).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Hughes, S.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Jouanin, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Kamp, M.

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Krauss, T. F.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

W. Whelan-Curtin, X. Yuan, and T. F. Krauss, 'Effects of disorder in photonic crystals,' presented at Proceedings of the PECS-VI Conference, Crete, 2005.

Kuramochi, E.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Labilloy, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Leuenberger, D.

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
[CrossRef]

López-Alonso, J. M.

J. M. Rico-García, J. M. López-Alonso, and J. Alda, Opt. Express 13, 3082 (2005), and references therein.
[CrossRef]

Moosbürger, J.

R. Ferrini, R. Houdré, H. Benisty, M. Qiu, and J. Moosbürger, J. Opt. Soc. Am. B 20, 469 (2003).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Mulot, M.

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

Notomi, M.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Qiu, M.

R. Ferrini, R. Houdré, H. Benisty, M. Qiu, and J. Moosbürger, J. Opt. Soc. Am. B 20, 469 (2003).
[CrossRef]

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

Ramunno, L.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Rico-García, J. M.

J. M. Rico-García, J. M. López-Alonso, and J. Alda, Opt. Express 13, 3082 (2005), and references therein.
[CrossRef]

Shinya, A.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
[CrossRef]

Sipe, J. E.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Skorobogatiy, M.

Smith, C. J. M.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
[CrossRef]

Talneau, A.

M. Skorobogatiy, G. Bégin, and A. Talneau, Opt. Express 13, 2487 (2005), and references therein.
[CrossRef] [PubMed]

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

Watanabe, T.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Weisbuch, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Whelan-Curtin, W.

W. Whelan-Curtin, X. Yuan, and T. F. Krauss, 'Effects of disorder in photonic crystals,' presented at Proceedings of the PECS-VI Conference, Crete, 2005.

Young, J. F.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Yuan, X.

W. Whelan-Curtin, X. Yuan, and T. F. Krauss, 'Effects of disorder in photonic crystals,' presented at Proceedings of the PECS-VI Conference, Crete, 2005.

Appl. Phys. Lett.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, C. Jouanin, and D. Cassagne, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, Appl. Phys. Lett. 85, 3998 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Nanotechnology

J. Moosbürger, M. Kamp, A. Forchel, R. Ferrini, D. Leuenberger, R. Houdré, S. Anand, and J. Bergreen, Nanotechnology 13, 341 (2002).
[CrossRef]

Opt. Express

J. M. Rico-García, J. M. López-Alonso, and J. Alda, Opt. Express 13, 3082 (2005), and references therein.
[CrossRef]

M. Skorobogatiy, G. Bégin, and A. Talneau, Opt. Express 13, 2487 (2005), and references therein.
[CrossRef] [PubMed]

Opt. Lett.

Photonics Nanostruct. Fundam. Appl.

D. Gerace and L. C. Andreani, Photonics Nanostruct. Fundam. Appl. 3, 120 (2005).
[CrossRef]

Phys. Rev. B

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
[CrossRef]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Phys. Rev. Lett.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Quantum Electron.

R. Ferrini, D. Leuenberger, M. Mulot, M. Qiu, J. Moosbürger, M. Kamp, A. Forchel, S. Anand, and R. Houdré, Quantum Electron. 38, 786 (2002).
[CrossRef]

Other

W. Whelan-Curtin, X. Yuan, and T. F. Krauss, 'Effects of disorder in photonic crystals,' presented at Proceedings of the PECS-VI Conference, Crete, 2005.

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Figures (3)

Fig. 1
Fig. 1

Schematic of (a) a planar PhC etched through a vertical waveguide with a refractive index profile n 1 n 2 n 1 (w, core thickness; h, hole depth); (b) and (c) In-plane positional (left) and geometrical (right) disorder effects (x and R, hole position and radius; S, area of the hole cross section). The perturbation with respect to the periodic lattice is represented by plugs of dielectric constant ± ( 1 n eff 2 ) that extend down to the hole bottom h with a cross-section area δ S .

Fig. 2
Fig. 2

(a) Measured (solid curves) and calculated (gray dotted curves) Γ M transmission spectra through eight-row-thick disordered PhC slabs for TE polarization: the value of the loss parameter ϵ is reported; the inset shows the top and edge SEM micrographs. (b) Periodic and (c) and disordered components of the top SEM image.

Fig. 3
Fig. 3

Disorder-induced losses ( ϵ dis ) as a function of the disorder parameter δ S S for GaAs-based PhCs. Circle: the disordered PhC shown in Fig. 2. Dotted lines and gray region: ϵ int ϵ fed and δ S S values, respectively, for the present fabrication techniques of GaAs-based PhCs.

Tables (1)

Tables Icon

Table 1 Calculated and Measured Radiation Losses

Equations (8)

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

ϵ = ϵ int + ϵ fed + ϵ dis ,
ϵ int n 2 Δ ϵ 2 λ 0 3 w S Γ 1 η ,
ϵ fed = 4 3 π 2 n 1 ( 1 n 1 2 ) 2 λ 0 3 2 L d S Γ ,
ϵ dis = 4 3 π 2 n eff ( 1 n eff 2 ) 2 λ 0 3 w eff S ( δ S S ) 2 ( 1 Γ ) η .
P rad = 4 3 π 3 c λ 0 4 ϵ 0 n eff ( 1 n eff 2 ) 2 S ( δ S S ) 2 S ψ 2 ( x , y ) d x d y [ h 0 ζ ( z ) d z ] 2 η ,
[ δ S ψ ( x , y ) d x d y ] 2 S ( δ S S ) 2 S ψ 2 ( x , y ) d x d y
P diss = 1 2 ϵ 0 2 π c λ 0 ϵ dis S ψ 2 ( x , y ) d x d y + ζ 2 ( z ) d z .
w eff [ h + ζ ( z ) d z ] 2 h + ζ 2 ( z ) d z [ + ζ ( z ) d z ] 2 + ζ 2 ( z ) d z .

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