Abstract

We study light transmission in two-dimensional photonic-crystal waveguides with embedded nonlinear defects. First, we derive effective discrete equations with long-range interaction for describing the waveguide modes and demonstrate that they provide a highly accurate generalization of the familiar tight-binding models that are employed, e.g., for the study of coupled-resonator optical waveguides. Using these equations, we investigate the properties of straight waveguides and waveguide bends with embedded linear and nonlinear defects. We emphasize the role of evanescent modes in the transmission properties of such waveguides and demonstrate the possibility of the nonlinearity-induced bistable (all-optical switcher) and unidirectional (optical diode) transmission. Additionally, we demonstrate adaptability of our approach for investigation of multimode waveguides by the example of the bound states in their constrictions.

© 2002 Optical Society of America

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  1. J. D. Joannoupoulos, R. B. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
  3. T. F. Krauss and R. M. De la Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999), and references therein.
    [CrossRef]
  4. See, e.g., K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999), and discussions therein.
    [CrossRef]
  5. S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168–1171 (1993).
    [CrossRef] [PubMed]
  6. S. John and N. Aközbek, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
    [CrossRef]
  7. S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E 62, 5777–5782 (2000).
    [CrossRef]
  8. S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
    [CrossRef] [PubMed]
  9. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001), http://epubs.osa.org//optics.express.
    [CrossRef] [PubMed]
  10. A. Mekis, S. H. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B 58, 4809–4817 (1998).
    [CrossRef]
  11. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
    [CrossRef] [PubMed]
  12. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
    [CrossRef] [PubMed]
  13. S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
    [CrossRef]
  14. S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
    [CrossRef]
  15. S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
    [CrossRef]
  16. T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
    [CrossRef]
  17. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [CrossRef]
  18. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387–400 (2000).
    [CrossRef]
  19. M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: a waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B 61, R11855–R11858 (2000).
    [CrossRef]
  20. E. Lidorikis, M. M. Sigalas, E. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
    [CrossRef]
  21. A. R. McGurn, “Green’s-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059–7064 (1996).
    [CrossRef]
  22. S. F. Mingaleev and Yu. S. Kivshar, “Effective equations for photonic crystal waveguides and circuits,” Opt. Lett. 27, 231–233 (2002).
    [CrossRef]
  23. F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
    [CrossRef] [PubMed]
  24. Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
    [CrossRef]
  25. E. Centero and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B 62, R7683–R7686 (2000).
    [CrossRef]
  26. M. Scalora, J. R. Dowling, C. M. Bowden, and M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
    [CrossRef]
  27. M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
    [CrossRef]

2002

2001

2000

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E 62, 5777–5782 (2000).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: a waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B 61, R11855–R11858 (2000).
[CrossRef]

E. Centero and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B 62, R7683–R7686 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387–400 (2000).
[CrossRef]

1999

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
[CrossRef]

T. F. Krauss and R. M. De la Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999), and references therein.
[CrossRef]

See, e.g., K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999), and discussions therein.
[CrossRef]

1998

S. John and N. Aközbek, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

A. Mekis, S. H. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B 58, 4809–4817 (1998).
[CrossRef]

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

E. Lidorikis, M. M. Sigalas, E. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

1996

A. R. McGurn, “Green’s-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059–7064 (1996).
[CrossRef]

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

1995

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

1994

M. Scalora, J. R. Dowling, C. M. Bowden, and M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

1993

S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168–1171 (1993).
[CrossRef] [PubMed]

1986

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Aközbek, N.

S. John and N. Aközbek, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168–1171 (1993).
[CrossRef] [PubMed]

Bayindir, M.

M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: a waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B 61, R11855–R11858 (2000).
[CrossRef]

Bloemer, M. J.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

M. Scalora, J. R. Dowling, C. M. Bowden, and M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Bowden, C. M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

M. Scalora, J. R. Dowling, C. M. Bowden, and M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Busch, K.

See, e.g., K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999), and discussions therein.
[CrossRef]

Centero, E.

E. Centero and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B 62, R7683–R7686 (2000).
[CrossRef]

Chan, C. T.

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Chow, E.

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

de Dood, M. J. A.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
[CrossRef]

De la Rue, R. M.

T. F. Krauss and R. M. De la Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999), and references therein.
[CrossRef]

Delyon, F.

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Dowling, J. P.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Dowling, J. R.

M. Scalora, J. R. Dowling, C. M. Bowden, and M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Economou, E.

E. Lidorikis, M. M. Sigalas, E. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Fan, S.

Fan, S. H.

A. Mekis, S. H. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B 58, 4809–4817 (1998).
[CrossRef]

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Felbacq, D.

E. Centero and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B 62, R7683–R7686 (2000).
[CrossRef]

Haus, H. A.

Hietala, V.

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

Ho, K. M.

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Joannopoulos, J. D.

S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001), http://epubs.osa.org//optics.express.
[CrossRef] [PubMed]

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

A. Mekis, S. H. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B 58, 4809–4817 (1998).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

John, S.

See, e.g., K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999), and discussions therein.
[CrossRef]

S. John and N. Aközbek, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. 71, 1168–1171 (1993).
[CrossRef] [PubMed]

Johnson, S. G.

Kivshar, Yu. S.

S. F. Mingaleev and Yu. S. Kivshar, “Effective equations for photonic crystal waveguides and circuits,” Opt. Lett. 27, 231–233 (2002).
[CrossRef]

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E 62, 5777–5782 (2000).
[CrossRef]

Krauss, T. F.

T. F. Krauss and R. M. De la Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999), and references therein.
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Lee, R. K.

Lévy, Y.-E.

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Li, Q.

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, E. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Lin, S. Y.

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

Manolatou, C.

McGurn, A. R.

A. R. McGurn, “Green’s-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059–7064 (1996).
[CrossRef]

Mekis, A.

A. Mekis, S. H. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B 58, 4809–4817 (1998).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Mingaleev, S. F.

S. F. Mingaleev and Yu. S. Kivshar, “Effective equations for photonic crystal waveguides and circuits,” Opt. Lett. 27, 231–233 (2002).
[CrossRef]

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E 62, 5777–5782 (2000).
[CrossRef]

Ozbay, E.

M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: a waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B 61, R11855–R11858 (2000).
[CrossRef]

Polman, A.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
[CrossRef]

Sammut, R. A.

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E 62, 5777–5782 (2000).
[CrossRef]

Scalora, M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

M. Scalora, J. R. Dowling, C. M. Bowden, and M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Scherer, A.

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, E. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Snoeks, E.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
[CrossRef]

Souillard, B.

F. Delyon, Y.-E. Lévy, and B. Souillard, “Nonperturbative bistability in periodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 (1986).
[CrossRef] [PubMed]

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, E. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[CrossRef]

Temelkuran, B.

M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: a waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B 61, R11855–R11858 (2000).
[CrossRef]

Tocci, M. D.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

van der Drift, E.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
[CrossRef]

Villeneuve, P. R.

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef] [PubMed]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Xu, Y.

Yariv, A.

Zijlstra, T.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
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Appl. Phys. Lett.

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M. Scalora, J. R. Dowling, C. M. Bowden, and M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

J. Opt. Soc. Am. B

J. Vac. Sci. Technol. B

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734–2739 (1999).
[CrossRef]

Opt. Express

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Phys. Rev. B

M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: a waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B 61, R11855–R11858 (2000).
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Figures (12)

Fig. 1
Fig. 1

Bandgap structure of the photonic crystal created by a square lattice of dielectric rods with r0=0.18a and 0=11.56; the bandgaps are shown crosshatched. The top right-hand schematic shows a cross-sectional view of the 2-D photonic crystal. The bottom right-hand schematic shows the corresponding Brillouin zone with the irreducible zone shaded.

Fig. 2
Fig. 2

Dispersion relation for a 2-D photonic-crystal waveguide (shown in the inset) as calculated by the supercell method9 (solid curve) and from approximate equations (10) and (11) for L=7 (dashed curve) and L=1 (dotted curve). The hatched areas are the projected band structure of a perfect 2-D crystal.

Fig. 3
Fig. 3

Same as for Fig. 2 but for two other types of waveguides better described by the tight-binding models. The solid curve represents results from supercell calculations; the dotted and the dashed curves represent results from the approximate equations.

Fig. 4
Fig. 4

Electric field En for five bound states supported by the multimode waveguide with the constriction shown in the center. The center of the constriction is located at n=0. The frequencies (ωa/2πc) of the bound states are (a) 0.389, (b) 0.391, (c) 0.392, (d) and (e) 0.402.

Fig. 5
Fig. 5

Wave vectors kp(ω) for ω=0.351 found by use of Eq. (11) for the waveguide shown in Fig. 2 (with a=1).

Fig. 6
Fig. 6

Transmission coefficients of an array of nonlinear defect rods calculated from Eqs. (8)–(13) with L=7 in the linear limit of very small |αt|2 (solid curves) and for nonlinear transmission when the output intensity is |αt|2=0.25 (dashed curves), for different numbers of the defects. We use nonlinear defect rods with the dielectric constant d(0)=7; they are marked by open circles on the diagrams on the right-hand side.

Fig. 7
Fig. 7

Bistability in the nonlinear transmission of an array of five nonlinear defect rods shown in Fig. 6(b).

Fig. 8
Fig. 8

Transmission coefficients of an asymmetric array of nonlinear defect rods calculated for the same parameters as in Fig. 6.

Fig. 9
Fig. 9

Nonlinear transmission of the optical diode for the forward (see top of Fig. 8) and the backward (see bottom of Fig. 8) directions at a light frequency of ω=0.326(2πc/a).

Fig. 10
Fig. 10

Reflection coefficients calculated by the finite-difference time-domain method (dashed curve; from Ref. 11) and from Eqs. (8)–(13) with L=7 (solid curves) and L=2 (dotted curve; only in the top plot) for different bend geometries.

Fig. 11
Fig. 11

Transmission of a waveguide bend with three embedded nonlinear defect rods in the linear (solid curve) and the nonlinear (dashed curve) regimes. The defect rods have a dielectric constant of d(0)=7, and they are marked by open circles.

Fig. 12
Fig. 12

Bistable nonlinear transmission through the waveguide bend shown in Fig. 11, for a light frequency of ω=0.351(2πc/a).

Equations (16)

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2+ωc2(x)E(x|ω)=0,
δ(x)=nd[E(x|ω)]f(x-xn),
E(x|ω)=ωc2d2yG(x, y|ω)δ(y)E(y|ω),
El(x)=ωlc2rdd2yG(x, y|ωl)df(y)El(y),
E(x|ω)=l,nψn(l)(ω)El(x-xn).
l,nλl,nl,nψn(l)=l,n,mdμl,n,ml,n(ω)ψn(l),
λl,nl,n=d2xEl(x-xn)El(x-xn),
μl,n,ml,n(ω)=ωc2d2xEl(x-xn)×d2yG(x, y|ω)f(y-xm)El(y-xn).
mMn,m(ω)Em(ω)=0,
Mn,m(ω)=d(Em)Jn,m(ω)-δn,m,
Jn,m(ω)=ωc2rdd2yG(xn, xm+y|ω)
T1, j(ω)=-ML-j(ω)ML(ω),forj=1, 2,,2L,
Tj,j+1=1,forj=1, 2,,2L-1.
Tˆ(ω)Φ p=exp{ikp(ω)}Φ p,
Emin=αiΦa-m1+αrΦa-m2+p=3L+1βpinΦa-mp,
Emout=αtΦm-b2+p=3L+1βpoutΦm-bp,

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