Abstract

High resolution images of planar photonic crystal (PC) optical components fabricated by e-beam lithography in various materials are analyzed to characterize statistical properties of common 2D geometrical imperfections. Our motivation is to attempt an intuitive, while rigorous statistical description of fabrication imperfections to provide a realistic input into theoretical modelling of PC device performance.

© 2005 Optical Society of America

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References

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  1. K.Y. Bliokh, Y.P. Bliokh, and V. D. Freilikher, “Resonances in one-dimensional disordered systems: localization of energy and resonant transmission,” J. Opt. Soc. Am. B 21, 113–120 (2004).
    [CrossRef]
  2. V.M. Apalkov, M.E. Raikh, and B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004).
    [CrossRef]
  3. E. Lidorikis and M.M. Sigalas, et al. “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61, 13458–13464 (2000).
    [CrossRef]
  4. A.A. Asatryan and P.A. Robinson, et al. “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 62, 5711–5720 (2000).
    [CrossRef]
  5. M.A. Kaliteevski and J.M. Martinez, et al. “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B 66, 113101 (2002).
    [CrossRef]
  6. K.-C. Kwan and X. Zhang, et al. “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).
    [CrossRef]
  7. B.Z. Steinberg, A. Boag, and R. Lisitsin, “Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy,” J. Opt. Soc. Am. A 20, 138 (2003).
    [CrossRef]
  8. B.C. Guptaa and Z. Yeb, “Disorder effects on the imaging of a negative refractive lens made by arrays of dielectric cylinders,” J. Appl. Phys. 94, 2173–2176 (2003).
    [CrossRef]
  9. S. Lan and K. Kanamoto, et al. “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B 67, 115208 (2003).
    [CrossRef]
  10. T. N. Langtry and A.A. Asatryan, et al. “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003).
    [CrossRef]
  11. A.G. Martyn and D. Hermann, et al. “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177183 (2003).
  12. N.A. Mortensen and M.D. Nielsen, et al. “Small-core photonic crystal fibres with weakly disordered air-hole claddings,” J. Opt. A: Pure Appl. Opt. 6, 221223 (2004).
    [CrossRef]
  13. M. Skorobogatiy, “Modelling the impact of imperfections in high index-contrast photonic waveguides,” Phys. Rev. E 70, 46609 (2004).
    [CrossRef]
  14. W. Bogaerts, P. Bienstman, and R. Baets, “Scattering at sidewall roughness in photonic crystal slabs,” Opt. Lett. 28, 689–691 (2003).
    [CrossRef] [PubMed]
  15. M.L. Povinelli and S.G. Johnson, et al. “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Lett. 84, 3639–3641 (2004).
    [CrossRef]
  16. S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995).
    [CrossRef]
  17. V. Yannopapas, A. Modinos, and N. Stefanou, “Anderson localization of light in inverted opals,” Phys. Rev. B 68, 193205 (2003).
    [CrossRef]
  18. D.J. Whitehouse, “Surface Characterization and Roughness Measurement in Engineering,” Photomechanics, Topics Appl. Phys. 77, 413461 (2000).
  19. D.J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc. Instn. Mech. Engrs. Part J 215, 207–210 (2001).
    [CrossRef]
  20. I. Arino and U. Kleist, et al. “Surface Texture Characterization of Injection-Molded Pigmented Plastics,” Polym. Eng. Sci. 44, 1615–1626 (2004).
    [CrossRef]
  21. Developed software and analysed PC images are available at http://www.photonics.phys.polymtl.ca/PolyFIT/
  22. A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003).
    [CrossRef]
  23. M. Mulot and S. Anand, et al. “Low-loss InP-based photonic crystal waveguides etched with Ar/Cl2 chemically assisted ion beam etching,” J. Vac. Sci. Technol. B21, 900–903 (2003).
  24. A. Xing and M. Davanco, et al. “Fabrication of InP-based two-dimensional photonic crystal membrane,” J. Vac. Sci. Technol. B 22, 70–73 (2004).
    [CrossRef]
  25. C. Monat and C. Seassal, et al. “Two-dimensional hexagonal-shaped microcavities formed in a two-dimensional photonic crystal on an InP membrane,” J. App. Phys. 93, 23–31 (2003).
    [CrossRef]
  26. P.E. Barclay and K. Srinivasan, et al. “Efficient input and output fiber coupling to a photonic crystal waveguide,” Opt. Lett. 29, 697–699 (2004).
    [CrossRef] [PubMed]
  27. H. Altuga and J. Vuckovic, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
    [CrossRef]
  28. M. Augustin and H.-J. Fuchs, et al. “High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,” Appl. Phys. Lett.84, (2004).
    [CrossRef]

2004 (9)

N.A. Mortensen and M.D. Nielsen, et al. “Small-core photonic crystal fibres with weakly disordered air-hole claddings,” J. Opt. A: Pure Appl. Opt. 6, 221223 (2004).
[CrossRef]

M. Skorobogatiy, “Modelling the impact of imperfections in high index-contrast photonic waveguides,” Phys. Rev. E 70, 46609 (2004).
[CrossRef]

M.L. Povinelli and S.G. Johnson, et al. “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Lett. 84, 3639–3641 (2004).
[CrossRef]

I. Arino and U. Kleist, et al. “Surface Texture Characterization of Injection-Molded Pigmented Plastics,” Polym. Eng. Sci. 44, 1615–1626 (2004).
[CrossRef]

A. Xing and M. Davanco, et al. “Fabrication of InP-based two-dimensional photonic crystal membrane,” J. Vac. Sci. Technol. B 22, 70–73 (2004).
[CrossRef]

H. Altuga and J. Vuckovic, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
[CrossRef]

K.Y. Bliokh, Y.P. Bliokh, and V. D. Freilikher, “Resonances in one-dimensional disordered systems: localization of energy and resonant transmission,” J. Opt. Soc. Am. B 21, 113–120 (2004).
[CrossRef]

V.M. Apalkov, M.E. Raikh, and B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004).
[CrossRef]

P.E. Barclay and K. Srinivasan, et al. “Efficient input and output fiber coupling to a photonic crystal waveguide,” Opt. Lett. 29, 697–699 (2004).
[CrossRef] [PubMed]

2003 (11)

B.Z. Steinberg, A. Boag, and R. Lisitsin, “Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy,” J. Opt. Soc. Am. A 20, 138 (2003).
[CrossRef]

W. Bogaerts, P. Bienstman, and R. Baets, “Scattering at sidewall roughness in photonic crystal slabs,” Opt. Lett. 28, 689–691 (2003).
[CrossRef] [PubMed]

C. Monat and C. Seassal, et al. “Two-dimensional hexagonal-shaped microcavities formed in a two-dimensional photonic crystal on an InP membrane,” J. App. Phys. 93, 23–31 (2003).
[CrossRef]

A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003).
[CrossRef]

M. Mulot and S. Anand, et al. “Low-loss InP-based photonic crystal waveguides etched with Ar/Cl2 chemically assisted ion beam etching,” J. Vac. Sci. Technol. B21, 900–903 (2003).

V. Yannopapas, A. Modinos, and N. Stefanou, “Anderson localization of light in inverted opals,” Phys. Rev. B 68, 193205 (2003).
[CrossRef]

K.-C. Kwan and X. Zhang, et al. “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).
[CrossRef]

B.C. Guptaa and Z. Yeb, “Disorder effects on the imaging of a negative refractive lens made by arrays of dielectric cylinders,” J. Appl. Phys. 94, 2173–2176 (2003).
[CrossRef]

S. Lan and K. Kanamoto, et al. “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B 67, 115208 (2003).
[CrossRef]

T. N. Langtry and A.A. Asatryan, et al. “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003).
[CrossRef]

A.G. Martyn and D. Hermann, et al. “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177183 (2003).

2002 (1)

M.A. Kaliteevski and J.M. Martinez, et al. “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B 66, 113101 (2002).
[CrossRef]

2001 (1)

D.J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc. Instn. Mech. Engrs. Part J 215, 207–210 (2001).
[CrossRef]

2000 (3)

D.J. Whitehouse, “Surface Characterization and Roughness Measurement in Engineering,” Photomechanics, Topics Appl. Phys. 77, 413461 (2000).

E. Lidorikis and M.M. Sigalas, et al. “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61, 13458–13464 (2000).
[CrossRef]

A.A. Asatryan and P.A. Robinson, et al. “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 62, 5711–5720 (2000).
[CrossRef]

1995 (1)

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995).
[CrossRef]

Altuga, H.

H. Altuga and J. Vuckovic, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
[CrossRef]

Anand, S.

M. Mulot and S. Anand, et al. “Low-loss InP-based photonic crystal waveguides etched with Ar/Cl2 chemically assisted ion beam etching,” J. Vac. Sci. Technol. B21, 900–903 (2003).

Apalkov, V.M.

Arino, I.

I. Arino and U. Kleist, et al. “Surface Texture Characterization of Injection-Molded Pigmented Plastics,” Polym. Eng. Sci. 44, 1615–1626 (2004).
[CrossRef]

Asatryan, A.A.

T. N. Langtry and A.A. Asatryan, et al. “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003).
[CrossRef]

A.A. Asatryan and P.A. Robinson, et al. “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 62, 5711–5720 (2000).
[CrossRef]

Augustin, M.

M. Augustin and H.-J. Fuchs, et al. “High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,” Appl. Phys. Lett.84, (2004).
[CrossRef]

Baets, R.

Barclay, P.E.

Bienstman, P.

Bliokh, K.Y.

Bliokh, Y.P.

Boag, A.

Bogaerts, W.

Davanco, M.

A. Xing and M. Davanco, et al. “Fabrication of InP-based two-dimensional photonic crystal membrane,” J. Vac. Sci. Technol. B 22, 70–73 (2004).
[CrossRef]

Fan, S.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995).
[CrossRef]

Freilikher, V. D.

Fuchs, H.-J.

M. Augustin and H.-J. Fuchs, et al. “High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,” Appl. Phys. Lett.84, (2004).
[CrossRef]

Guptaa, B.C.

B.C. Guptaa and Z. Yeb, “Disorder effects on the imaging of a negative refractive lens made by arrays of dielectric cylinders,” J. Appl. Phys. 94, 2173–2176 (2003).
[CrossRef]

Hermann, D.

A.G. Martyn and D. Hermann, et al. “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177183 (2003).

Joannopoulos, J.D.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995).
[CrossRef]

Johnson, S.G.

M.L. Povinelli and S.G. Johnson, et al. “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Lett. 84, 3639–3641 (2004).
[CrossRef]

Kaliteevski, M.A.

M.A. Kaliteevski and J.M. Martinez, et al. “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B 66, 113101 (2002).
[CrossRef]

Kanamoto, K.

S. Lan and K. Kanamoto, et al. “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B 67, 115208 (2003).
[CrossRef]

Kleist, U.

I. Arino and U. Kleist, et al. “Surface Texture Characterization of Injection-Molded Pigmented Plastics,” Polym. Eng. Sci. 44, 1615–1626 (2004).
[CrossRef]

Kwan, K.-C.

K.-C. Kwan and X. Zhang, et al. “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).
[CrossRef]

Lan, S.

S. Lan and K. Kanamoto, et al. “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B 67, 115208 (2003).
[CrossRef]

Langtry, T. N.

T. N. Langtry and A.A. Asatryan, et al. “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003).
[CrossRef]

Lidorikis, E.

E. Lidorikis and M.M. Sigalas, et al. “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61, 13458–13464 (2000).
[CrossRef]

Lisitsin, R.

Martinez, J.M.

M.A. Kaliteevski and J.M. Martinez, et al. “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B 66, 113101 (2002).
[CrossRef]

Martyn, A.G.

A.G. Martyn and D. Hermann, et al. “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177183 (2003).

Modinos, A.

V. Yannopapas, A. Modinos, and N. Stefanou, “Anderson localization of light in inverted opals,” Phys. Rev. B 68, 193205 (2003).
[CrossRef]

Monat, C.

C. Monat and C. Seassal, et al. “Two-dimensional hexagonal-shaped microcavities formed in a two-dimensional photonic crystal on an InP membrane,” J. App. Phys. 93, 23–31 (2003).
[CrossRef]

Mortensen, N.A.

N.A. Mortensen and M.D. Nielsen, et al. “Small-core photonic crystal fibres with weakly disordered air-hole claddings,” J. Opt. A: Pure Appl. Opt. 6, 221223 (2004).
[CrossRef]

Mulot, M.

A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003).
[CrossRef]

M. Mulot and S. Anand, et al. “Low-loss InP-based photonic crystal waveguides etched with Ar/Cl2 chemically assisted ion beam etching,” J. Vac. Sci. Technol. B21, 900–903 (2003).

Nielsen, M.D.

N.A. Mortensen and M.D. Nielsen, et al. “Small-core photonic crystal fibres with weakly disordered air-hole claddings,” J. Opt. A: Pure Appl. Opt. 6, 221223 (2004).
[CrossRef]

Povinelli, M.L.

M.L. Povinelli and S.G. Johnson, et al. “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Lett. 84, 3639–3641 (2004).
[CrossRef]

Raikh, M.E.

Robinson, P.A.

A.A. Asatryan and P.A. Robinson, et al. “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 62, 5711–5720 (2000).
[CrossRef]

Seassal, C.

C. Monat and C. Seassal, et al. “Two-dimensional hexagonal-shaped microcavities formed in a two-dimensional photonic crystal on an InP membrane,” J. App. Phys. 93, 23–31 (2003).
[CrossRef]

Shapiro, B.

Sigalas, M.M.

E. Lidorikis and M.M. Sigalas, et al. “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61, 13458–13464 (2000).
[CrossRef]

Skorobogatiy, M.

M. Skorobogatiy, “Modelling the impact of imperfections in high index-contrast photonic waveguides,” Phys. Rev. E 70, 46609 (2004).
[CrossRef]

Srinivasan, K.

Stefanou, N.

V. Yannopapas, A. Modinos, and N. Stefanou, “Anderson localization of light in inverted opals,” Phys. Rev. B 68, 193205 (2003).
[CrossRef]

Steinberg, B.Z.

Talneau, A.

A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003).
[CrossRef]

Villeneuve, P.R.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995).
[CrossRef]

Vuckovic, J.

H. Altuga and J. Vuckovic, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
[CrossRef]

Whitehouse, D.J.

D.J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc. Instn. Mech. Engrs. Part J 215, 207–210 (2001).
[CrossRef]

D.J. Whitehouse, “Surface Characterization and Roughness Measurement in Engineering,” Photomechanics, Topics Appl. Phys. 77, 413461 (2000).

Xing, A.

A. Xing and M. Davanco, et al. “Fabrication of InP-based two-dimensional photonic crystal membrane,” J. Vac. Sci. Technol. B 22, 70–73 (2004).
[CrossRef]

Yannopapas, V.

V. Yannopapas, A. Modinos, and N. Stefanou, “Anderson localization of light in inverted opals,” Phys. Rev. B 68, 193205 (2003).
[CrossRef]

Yeb, Z.

B.C. Guptaa and Z. Yeb, “Disorder effects on the imaging of a negative refractive lens made by arrays of dielectric cylinders,” J. Appl. Phys. 94, 2173–2176 (2003).
[CrossRef]

Zhang, X.

K.-C. Kwan and X. Zhang, et al. “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).
[CrossRef]

Appl. Phys. Lett. (4)

K.-C. Kwan and X. Zhang, et al. “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).
[CrossRef]

M.L. Povinelli and S.G. Johnson, et al. “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Lett. 84, 3639–3641 (2004).
[CrossRef]

A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003).
[CrossRef]

H. Altuga and J. Vuckovic, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
[CrossRef]

J. App. Phys. (1)

C. Monat and C. Seassal, et al. “Two-dimensional hexagonal-shaped microcavities formed in a two-dimensional photonic crystal on an InP membrane,” J. App. Phys. 93, 23–31 (2003).
[CrossRef]

J. Appl. Phys. (2)

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995).
[CrossRef]

B.C. Guptaa and Z. Yeb, “Disorder effects on the imaging of a negative refractive lens made by arrays of dielectric cylinders,” J. Appl. Phys. 94, 2173–2176 (2003).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

N.A. Mortensen and M.D. Nielsen, et al. “Small-core photonic crystal fibres with weakly disordered air-hole claddings,” J. Opt. A: Pure Appl. Opt. 6, 221223 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

J. Vac. Sci. Technol. (1)

M. Mulot and S. Anand, et al. “Low-loss InP-based photonic crystal waveguides etched with Ar/Cl2 chemically assisted ion beam etching,” J. Vac. Sci. Technol. B21, 900–903 (2003).

J. Vac. Sci. Technol. B (1)

A. Xing and M. Davanco, et al. “Fabrication of InP-based two-dimensional photonic crystal membrane,” J. Vac. Sci. Technol. B 22, 70–73 (2004).
[CrossRef]

Nanotechnology (1)

A.G. Martyn and D. Hermann, et al. “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177183 (2003).

Opt. Lett. (2)

Photomechanics, Topics Appl. Phys. (1)

D.J. Whitehouse, “Surface Characterization and Roughness Measurement in Engineering,” Photomechanics, Topics Appl. Phys. 77, 413461 (2000).

Phys. Rev. B (4)

V. Yannopapas, A. Modinos, and N. Stefanou, “Anderson localization of light in inverted opals,” Phys. Rev. B 68, 193205 (2003).
[CrossRef]

M.A. Kaliteevski and J.M. Martinez, et al. “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B 66, 113101 (2002).
[CrossRef]

E. Lidorikis and M.M. Sigalas, et al. “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61, 13458–13464 (2000).
[CrossRef]

S. Lan and K. Kanamoto, et al. “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B 67, 115208 (2003).
[CrossRef]

Phys. Rev. E (3)

T. N. Langtry and A.A. Asatryan, et al. “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003).
[CrossRef]

A.A. Asatryan and P.A. Robinson, et al. “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 62, 5711–5720 (2000).
[CrossRef]

M. Skorobogatiy, “Modelling the impact of imperfections in high index-contrast photonic waveguides,” Phys. Rev. E 70, 46609 (2004).
[CrossRef]

Polym. Eng. Sci. (1)

I. Arino and U. Kleist, et al. “Surface Texture Characterization of Injection-Molded Pigmented Plastics,” Polym. Eng. Sci. 44, 1615–1626 (2004).
[CrossRef]

Proc. Instn. Mech. Engrs. Part J (1)

D.J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc. Instn. Mech. Engrs. Part J 215, 207–210 (2001).
[CrossRef]

Other (2)

Developed software and analysed PC images are available at http://www.photonics.phys.polymtl.ca/PolyFIT/

M. Augustin and H.-J. Fuchs, et al. “High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,” Appl. Phys. Lett.84, (2004).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

(a) Image of a hole together with a detected edge. (b) Shape of a rugged edge is fitted with Fourier series in θ. Smooth curve is an M=1 circle fit. (c) On a scale <2nm hole edge can not be represented by a single valued analytical function rfitM (θ). (d) Edge roughness is self-similar on very different scales suggesting fractal description.

Fig. 2.
Fig. 2.

(a,c)Probability density distribution of fit error for different number of angular momenta components M in a fit. (b) RMS of fit error decreases slowly as the number of angular momenta components M in a fit increases, suggesting that there is no simple coarse description of a feature shape. (c) RMS of fit error decreases dramatically when ellipticity M=2 of a feature is included in a fit, suggesting ellipticity as a dominant coarse parameter.

Fig. 3.
Fig. 3.

(a) “Height to height” correlation function and (b) auto-correlation function of an edge deviation from smooth fits with M angular components.

Fig. 4.
Fig. 4.

(a) Power spectral density (blue). Linear fit is over 2 decades starting from the largest length scale. (b) RMS of a fit error (blue). Linear fit spans the lowest angular momenta starting with M=1. (c) Power spectral density (blue). Linear fit is over 1 decade in the interval 30nm≳λ≳200nm (d) RMS of a fit error (blue). Linear fit is in the range 4<M≲40. In red are the statistical functions of a noise level due to finite resolution of an image.

Fig. 5.
Fig. 5.

(a)PC lattice of holes with 2 missing rows. Vertices of a fitted perfectly periodic underlying lattice are shows as white dots. (b)PDDs of hole center deviations from the vertices of a perfect lattice along 2 principal directions (solids) together with Gaussian fits (dotted lines): perpendicular to the waveguide σ1 (blue), and parallel to the waveguide σ2 (red). (c) RMS deviations σ1,2 (along 2 principle directions) of hole centers from an underlying lattice against the number of features in a fit. Features in a fit are included one by one, row by row starting from the upper left corner of an image.

Fig. 6.
Fig. 6.

(a) Uniform square PC lattice [26]. (b) σ1,2 as a function of the number of features in a fit. Distribution of feature centers around the vertices of an underlying perfect lattice is isotropic. (c) Triangular PC lattice with a waveguide and a bend [28]. (d) σ1,2 as a function of the number of features in a fit. Distribution of feature centers around the vertices of an underlying perfect lattice is anisotropic.

Fig. 7.
Fig. 7.

(a) Image of a hole with a moderate contrast and a high noise level. Insert: histogram of pixel values. Hole edges are detected with: (b) tol=0.37 (c) tol=0.40 (d) tol=0.43

Tables (4)

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Table 1. Parameterization of features in Fig. 1(a), InP/InGaAsP/InP [22].

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Table 2. Parameterization of features in Fig. 2(c), Air/Si membranes [26].

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Table 3. Parameterization of features in Fig. 6(a), Air/Si membranes [26].

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Table 4. Ranges of statistical parameters over various PC lattices [21].

Equations (23)

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Q edge M = 1 N edge i = 1 N edge ( r fit M ( θ i ) r edge ( θ i ) ) 2 ,
r fit M ( θ i ) = R 0 + m = 2 M ( A m M Sin ( m θ i ) + B m M Cos ( m θ i ) ) .
R = R a v ± δ R a v R a v = 1 N f n = 1 N f R 0 n δ R a v 2 = 1 N f n = 1 N f ( R 0 n R a v ) 2
σ 2 ( M ) = n = 1 N f N edge n σ n 2 ( M ) n = 1 N f N edge n
[ C , Γ , S ] M ( λ ) = 1 N f n = 1 N f [ C , Γ , S ] n M ( λ ) .
δ r M = r fit n , M ( θ i ) r edge n ( θ i ) ,
P ( δ r M ) = 1 2 π σ ( M ) exp ( δ r M ) 2 2 σ 2 ( M ) .
δ n M ( θ i ) = r fit n , M ( θ i ) r edge n ( θ i ) ,
f ( θ + ε ) f ( θ ) ε H , ε 0 ,
C n M ( λ ) = ( δ n M ( θ + λ R 0 n ) δ n M ( θ ) ) 2 θ = 1 2 π 0 2 π d θ ( δ n M ( θ + λ R 0 n ) δ n M ( θ ) ) 2 .
C n M ( λ ) = 2 ( σ n 2 ( M ) Γ n M ( λ ) ) ,
Γ n M ( λ ) = δ n M ( θ + λ R 0 n ) δ n M ( θ ) θ δ n N ( θ ) θ 2 .
C n M ( λ ) λ 0 λ 2 H ; C n M ( λ ) λ > λ n c M 2 σ n 2 ( M ) Γ n M ( λ ) λ > λ n c M 0 ; ( σ n 2 ( M ) Γ n M ( λ ) ) λ 0 λ 2 H .
C n M ( λ ) = 2 σ n 2 ( M ) ( 1 exp ( ( λ λ n c M ) 2 H ) )
Γ n M ( λ ) = σ n 2 ( M ) exp ( ( λ λ n c M ) 2 H ) .
δ n 1 ( θ ) = r edge n ( θ ) R 0 n = m = 2 N edge n ( A m Sin ( m θ ) + B m Cos ( m θ ) ) ,
Γ n 1 ( λ ) = 1 2 m = 2 N edge n ( A m 2 + B m 2 ) cos ( m λ R 0 n ) .
S n 1 ( λ m ) = 1 2 π R 0 n 0 2 π R 0 n d λ ˜ Γ n 1 ( λ ˜ ) exp ( i 2 π λ m λ ˜ ) = 1 4 ( A m 2 + B m 2 ) ,
S n 1 ( λ m ) = 1 4 ( A m 2 + B m 2 ) λ m 0 λ m 1 + 2 H ,
δ n M ( θ ) = m = M + 1 N edge n ( A m Sin ( m θ ) + B m Cos ( m θ ) ) .
σ n 2 ( M ) = ( δ n M ( θ ) ) 2 θ = 1 2 m = M + 1 N edge n ( A m 2 + B m 2 ) M H .
Q lat = 1 N f n = 1 N f ( r ̅ 0 n j 1 n a ̅ 1 j 2 n a ̅ 2 ) 2 ,
P ( δ ̅ c ) = 1 2 π σ 1 σ 2 exp ( ( δ c x δ c y ) T R T ( 1 2 σ 1 2 0 0 1 2 σ 2 2 ) R ( δ c x δ c y ) ) ,

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