Abstract

A heuristic formalism is developed for efficiently determining the specular reflectivity spectrum of two-dimensionally textured planar waveguides. The formalism is based on a Green’s function approach wherein the electric fields are assumed to vary little over the thickness of the textured part of the waveguide. Its accuracy, when the thickness of the textured region is much smaller than the wavelength of relevant radiation, is verified by comparison with a much less efficient, exact finite difference solution of Maxwell’s equations. In addition to its numerical efficiency, the formalism provides an intuitive explanation of Fano-like features evident in the specular reflectivity spectrum when the incident radiation is phase matched to excite leaky electromagnetic modes attached to the waveguide. By associating various Fourier components of the scattered field with bare slab modes, the dispersion, unique polarization properties, and lifetimes of these Fano-like features are explained in terms of photonic eigenmodes that reveal the renormalization of the slab modes due to interaction with the two-dimensional grating. An application of the formalism, in the analysis of polarization-insensitive notch filters, is also discussed.

© 2001 Optical Society of America

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  1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  3. J. D. Joannopoulos, P. R. Villeniuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
    [CrossRef]
  4. K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
    [CrossRef]
  5. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
    [CrossRef] [PubMed]
  6. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
    [CrossRef] [PubMed]
  7. M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
    [CrossRef]
  8. V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
    [CrossRef]
  9. V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
    [CrossRef]
  10. V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
    [CrossRef]
  11. S. M. Norton, T. Erdogan, G. Michael Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
    [CrossRef]
  12. S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
    [CrossRef]
  13. T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
    [CrossRef]
  14. J. F. Young, P. Paddon, V. Pacradouni, T. Tiedje, S. Johnson, “Photonic lattices in semiconductor waveguides,” in Future Trends in Microelectronics, S. Luryi, J. Xu, A. Zaslavsky, eds. (Wiley, Toronto, 1999), pp. 423–432.
  15. D. M. Whittaker, I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
    [CrossRef]
  16. S. Peng, G. M. Morris, “Resonant Scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  17. D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
    [CrossRef]
  18. P. Paddon, J. F. Young, “Simple approach to coupling in textured planar waveguides,” Opt. Lett. 23, 1529–1531 (1998).
    [CrossRef]
  19. P. Paddon, J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B 61, 2090–2101 (2000).
    [CrossRef]
  20. A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  21. W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
    [CrossRef]
  22. L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
    [CrossRef]
  23. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
    [CrossRef]
  24. S. Peng, G. M. Morris, “Experimental investigation of resonant grating filters based on two-dimensional gratings,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds, Proc. SPIE2689, 90–94 (1996).
    [CrossRef]
  25. V. Pacradouni, A. R. Cowan, J. Mandeville, P. Paddon, J. F. Young, “Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory,” post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, September 26–30, 1999.
  26. A. R. Cowan, “Periodically textured planar waveguides,” M. S. thesis (University of British Columbia, Vancouver, B.C., Canada, 2000), p. 32–36.

2000 (2)

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

P. Paddon, J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B 61, 2090–2101 (2000).
[CrossRef]

1999 (4)

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

D. M. Whittaker, I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

1998 (1)

1997 (5)

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

J. D. Joannopoulos, P. R. Villeniuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

S. M. Norton, T. Erdogan, G. Michael Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
[CrossRef]

T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
[CrossRef]

S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
[CrossRef]

1996 (2)

S. Peng, G. M. Morris, “Resonant Scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

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

1994 (1)

K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
[CrossRef]

1991 (1)

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

1988 (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

1987 (3)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

1977 (1)

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

1973 (1)

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Agi, K.

K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
[CrossRef]

Astratov, V. N.

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

Atkin, D. M.

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

Birks, T. A.

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

Brommer, K. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Brown, E. R.

K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Busch, A.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Calshaw, I. S.

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

Cowan, A. R.

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

V. Pacradouni, A. R. Cowan, J. Mandeville, P. Paddon, J. F. Young, “Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory,” post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, September 26–30, 1999.

A. R. Cowan, “Periodically textured planar waveguides,” M. S. thesis (University of British Columbia, Vancouver, B.C., Canada, 2000), p. 32–36.

Culshaw, I. S.

D. M. Whittaker, I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

De La Rue, R. M.

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

Dill, C.

K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
[CrossRef]

Erdogan, T.

Fan, S.

J. D. Joannopoulos, P. R. Villeniuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Hall, D. G.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeniuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S.

J. F. Young, P. Paddon, V. Pacradouni, T. Tiedje, S. Johnson, “Photonic lattices in semiconductor waveguides,” in Future Trends in Microelectronics, S. Luryi, J. Xu, A. Zaslavsky, eds. (Wiley, Toronto, 1999), pp. 423–432.

Johnson, S. R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Kanskar, M.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Krauss, T. F.

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Mackenzie, J.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Magnusson, R.

Malloy, K. J.

K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
[CrossRef]

Mandeville, J.

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

V. Pacradouni, A. R. Cowan, J. Mandeville, P. Paddon, J. F. Young, “Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory,” post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, September 26–30, 1999.

McMahon, O. B.

K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
[CrossRef]

Meade, R. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Morin, R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Morris, G. M.

S. Peng, G. M. Morris, “Resonant Scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

S. Peng, G. M. Morris, “Experimental investigation of resonant grating filters based on two-dimensional gratings,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds, Proc. SPIE2689, 90–94 (1996).
[CrossRef]

Morris, G. Michael

Norton, S. M.

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Pacradouni, V.

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

V. Pacradouni, A. R. Cowan, J. Mandeville, P. Paddon, J. F. Young, “Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory,” post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, September 26–30, 1999.

J. F. Young, P. Paddon, V. Pacradouni, T. Tiedje, S. Johnson, “Photonic lattices in semiconductor waveguides,” in Future Trends in Microelectronics, S. Luryi, J. Xu, A. Zaslavsky, eds. (Wiley, Toronto, 1999), pp. 423–432.

Paddon, P.

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

P. Paddon, J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B 61, 2090–2101 (2000).
[CrossRef]

P. Paddon, J. F. Young, “Simple approach to coupling in textured planar waveguides,” Opt. Lett. 23, 1529–1531 (1998).
[CrossRef]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

V. Pacradouni, A. R. Cowan, J. Mandeville, P. Paddon, J. F. Young, “Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory,” post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, September 26–30, 1999.

J. F. Young, P. Paddon, V. Pacradouni, T. Tiedje, S. Johnson, “Photonic lattices in semiconductor waveguides,” in Future Trends in Microelectronics, S. Luryi, J. Xu, A. Zaslavsky, eds. (Wiley, Toronto, 1999), pp. 423–432.

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Peng, S.

S. Peng, G. M. Morris, “Resonant Scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

S. Peng, G. M. Morris, “Experimental investigation of resonant grating filters based on two-dimensional gratings,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds, Proc. SPIE2689, 90–94 (1996).
[CrossRef]

Rappe, A. M.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Roberts, P. J.

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

Russell, P. St. J.

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

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Sipe, J. E.

Skolnick, M. S.

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

Stevenson, R. M.

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Tamir, T.

Tibuleac, S.

Tiedje, T.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

J. F. Young, P. Paddon, V. Pacradouni, T. Tiedje, S. Johnson, “Photonic lattices in semiconductor waveguides,” in Future Trends in Microelectronics, S. Luryi, J. Xu, A. Zaslavsky, eds. (Wiley, Toronto, 1999), pp. 423–432.

Villeniuve, P. R.

J. D. Joannopoulos, P. R. Villeniuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Whittaker, D. M.

V. N. Astratov, I. S. Calshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2056 (1999).
[CrossRef]

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

D. M. Whittaker, I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Young, J. F.

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

P. Paddon, J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B 61, 2090–2101 (2000).
[CrossRef]

P. Paddon, J. F. Young, “Simple approach to coupling in textured planar waveguides,” Opt. Lett. 23, 1529–1531 (1998).
[CrossRef]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

V. Pacradouni, A. R. Cowan, J. Mandeville, P. Paddon, J. F. Young, “Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory,” post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, September 26–30, 1999.

J. F. Young, P. Paddon, V. Pacradouni, T. Tiedje, S. Johnson, “Photonic lattices in semiconductor waveguides,” in Future Trends in Microelectronics, S. Luryi, J. Xu, A. Zaslavsky, eds. (Wiley, Toronto, 1999), pp. 423–432.

Zhang, S.

Appl. Phys. Lett. (1)

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, T. Tiedje, “Observation of leaky slab modes in air–bridge semiconductor waveguides with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Electron. Lett. (1)

K. Agi, E. R. Brown, O. B. McMahon, C. Dill, K. J. Malloy, “Design of ultrawideband photonic broadband antenna applications,” Electron. Lett. 30, 2166–2167 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

J. Lightwave Technol. (2)

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (4)

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

Nature (1)

J. D. Joannopoulos, P. R. Villeniuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (4)

D. M. Whittaker, I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

P. Paddon, J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B 61, 2090–2101 (2000).
[CrossRef]

V. N. Astratov, D. M. Whittaker, I. S. Calshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Science (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Other (4)

J. F. Young, P. Paddon, V. Pacradouni, T. Tiedje, S. Johnson, “Photonic lattices in semiconductor waveguides,” in Future Trends in Microelectronics, S. Luryi, J. Xu, A. Zaslavsky, eds. (Wiley, Toronto, 1999), pp. 423–432.

S. Peng, G. M. Morris, “Experimental investigation of resonant grating filters based on two-dimensional gratings,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds, Proc. SPIE2689, 90–94 (1996).
[CrossRef]

V. Pacradouni, A. R. Cowan, J. Mandeville, P. Paddon, J. F. Young, “Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory,” post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, September 26–30, 1999.

A. R. Cowan, “Periodically textured planar waveguides,” M. S. thesis (University of British Columbia, Vancouver, B.C., Canada, 2000), p. 32–36.

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Figures (8)

Fig. 1
Fig. 1

Schematic of a planar waveguide textured with a 2D planar grating. The grating has thickness tg, pitch Λ. The dielectric constant of the cylinders is ϵg, and they are embedded in a layer of the material that has a dielectric constant ϵc.

Fig. 2
Fig. 2

Reflectivity spectrum at β/βg=0.01xˆ for structure 1 described in the text. Solid curve, s-polarized radiation; dotted curve, p-polarized radiation. Each resonance is due to coupling into the radiative component of the renormalized slab modes with (1) the lowest-order TE-like modes, (2) the next-higher-order TE-like modes, and (3) the lowest-order TM-like modes.

Fig. 3
Fig. 3

Schematic plot of the band structure across the first Brillouin zone in the X and M directions for a weak square grating. This plot shows the kinematic coupling (zone folding) of all modes to the first Brillouin zone. The renormalization effects of the grating, which lead to avoided crossings, and zero slope for the bands at the Brillouin zone boundaries are not described by these purely kinematic effects. The insets show the nine Fourier components and the wave vectors of the eigenmode components with detuning in the X and M directions.

Fig. 4
Fig. 4

Reflectivity spectra for increasing |β/βg| away from zone center for a square grating (structure 1) with detuning in (a) the X and (b) the M symmetry directions. Solid curve, s-polarized radiation; dotted curve, p-polarized radiation. As resonances approach zone center, their width approaches zero or they become degenerate with opposite polarizations. The different dispersion and lifetime properties of the eigenmodes are evident.

Fig. 5
Fig. 5

Dispersion diagram for a square grating in the X direction within the first Brillouin zone. Solid curves, s-polarized eigenmodes; dotted curve, p-polarized eigenmodes. The symbols represent the location of peaks in the first-order diffracted spectra, peaks that correspond to the true resonance frequency of the Fano-like features in Fig. 4(a).

Fig. 6
Fig. 6

Reflectivity spectrum near the Brillouin zone boundary (β/βg=0.485xˆ) for the square grating (structure 1). The true bound modes appear as poles in the specular reflectivity spectrum. Of the four p-polarized modes, the two at lower ω˜ are TE-like modes and the upper two are the lowest-order TM-like modes.

Fig. 7
Fig. 7

Reflectivity spectra for structure 1 as calculated with an exact integration of Maxwell’s equations (lower plot) and with the approach described in this paper (upper plot). The resonance widths and shape are in excellent agreement. The gap width agrees to within 7%, and the center of the gap agrees to within 0.6%. The inset shows the location of the upper and lower edges of the second-order gap in a 1D textured symmetric waveguide of thickness tg, ϵs=12.25, ϵg=1.0, and a filling fraction of 25%, as calculated with the Green’s function (dashed curves) and exact (solid curves) methods.

Fig. 8
Fig. 8

Linewidth of the notch filter shown in the inset as it varies with the location of the grating in the slab. In this structure the linewidth can be altered by a factor of 6 by just shifting the location of the grating by 70 nm. The inset shows the reflectivity spectra of a polarization-insensitive, normally incident notch filter based on a honeycomb grating. See text for the structure’s parameters. The reflectivity spectrum is independent of the incident polarization.

Equations (39)

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·D=0,
×B+iωc D=0,
·B=0,
×E-iωc B=0.
D=E+4πPtot,
Ptot(ω; r)=χ(ω; r)E(ω; r).
χ(ω; r)=χs(ω; z)+Δχg(ω; z, ρ)
·[ϵs(ω; z)E(ω; r)]=-4π·ΔPg(ω; r),
×B(ω; r)+iω˜ϵs(ω; z)E(ω; r)=-4πiω˜ΔPg(ω; r),
·B(ω; r)=0,
×E(ω; r)-iω˜B(ω; r)=0,
EUHShom(ω; ρ, z)={exp[-iwo(β)z]1+exp[+iwo(β)z]rwg(β)}·Einc exp[iβ·ρ-iωt]θ(z-zt)
ϵc(ω)·E(ω; ρ, z)=-4π·ΔPg(ω; ρ, z),
×B(ω; ρ, z)+iω˜ϵc(ω)E(ω; ρ, z)
=-4πiω˜ΔPg(ω; ρ, z),
·B(ω; ρ, z)=0,
×E(ω; ρ, z)-iω˜B(ω; ρ, z)=0.
E(ω, κ; z)=-dzgd(κ; z, z)·ΔPg(ω, κ; z),
gd(κ; z, z)=2πiω˜2wc(κ){θ(z-z)exp[iwc(κ)(z-z)]
×[sˆ(κ)sˆ(κ)+pˆc+(κ)pˆc+(κ)]
+θ(z-z)exp[-iwc(κ)(z-z)]
×[sˆ(κ)sˆ(κ)+pˆc-(κ)pˆc-(κ)]}
-4πϵcδ(z-z)zˆzˆ,
sˆ(κ)=κˆ×zˆ,
pˆc±(κ)=κzˆ±wc(κ)κˆω˜ϵc.
gc(κ, z, z)=2πiω˜2wc(κ) θ(z-z)exp[iwc(κ)(z-z)]+θ(z-z)exp[-iwc(κ)(z-z)]+(rsup exp[-iwc(κ)(z+z-L)]+rsdown exp[iwc(κ)(z+z+L)]+rsuprsdown{exp[iwc(κ)(z-z+2L)]+exp[-iwc(κ)(z-z-2L)]}) 1Dssˆ(κ)sˆ(κ)+θ(z-z)exp[iwc(κ)(z-z)]+rpuprpdown exp[iwc(κ)(z-z+2L)]Dppˆc+(κ)pˆc+(κ)+θ(z-z)exp[-iwc(κ)(z-z)]+rpuprpdown exp[-iwc(κ)(z-z-2L)]Dppˆc-(κ)pˆc-(κ)+1Dp [rpup exp[-iwc(κ)(z+z-L)]pˆc-(κ)pˆc+(κ)+rpdown exp[iwc(κ)(z+z+L)]pˆc+(κ)pˆc-(κ)]-4πϵcδ(z-z)zˆzˆ,
Ds,p=1-rs,puprs,pdown exp[iwc(κ)2L]
Δχg(ω, r)=mχGm(ω, z)exp(iGm·ρ),
Ec(ω, β; z)=Echom(ω, β; z)+-L/2L/2dzgc(β; z, z)×mχGm(ω; z)Ec(ω, β-Gm; z),
EUHS(ω, β; zabove)=EUHShom(ω, β; zabove)+-L/2L/2dzgUHS(β; zabove, z)·ΔPg(ω, β; z),
gUHS(β, zabove, z)=2πiω˜2wc(β) exp[iwo(β)(zabove-zt)]×exp[iwc(β)(L/2-z)]×tsupDs{1+rsdown exp[iwc(β)(L+2z)]}sˆ(β)sˆ(β)+tpupDp{pˆc+(β)p˜c+(β)+pˆc+(β)pˆc-(β)rpdown×exp[iwc(β)(L+2z)]},
E(c)n(z)=E(c)nhom(z)+-L/2L/2dzg(c)n(z, z)mχnm(z)E(c)m(z),
En(zo)=Enhom(zo)+In(zo)mχnmEm(zo),
In(zo)=zo-tg/2zo+tg/2g(c)n(zo, z)dz.
E(zo)=Ehom(zo)+M(zo)·E(zo).
E(zo)=(1-M(zo))-1Ehom(zo).
E(zt)=Ehom(zt)+N(zt)E(zo).
E(zt)=Ehom(zt)+N(zt)(1-M(zo))-1Ehom(zo),
ESR(zt)={N(zt)[1-M(zo)]-1Ehom(zo)}specular+rwg(zt)Einc(zt).

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