Abstract

Specular reflectivity spectra of plane waves incident upon two-dimensional waveguide-based photonic crystals are rigorously calculated by use of the linear and the third-order nonlinear susceptibilities of the semiconductor core layer. The Fano-like features associated with coupling to leaky photonic eigenstates that are attached to the porous slab are shifted and distorted at high intensities. Although some of this nonlinear behavior is qualitatively similar to that observed in simple Fabry–Perot cavities, there are striking differences. The main difference is that one can engineer the Q values and the in-plane dispersion of the microcavity modes associated with the leaky eigenstates of the photonic crystal over a wide range by varying the properties of the etched texture. Examples are given that demonstrate bistable behavior and intensity-dependent reflectivities that can vary from zero to unity. Both degenerate (single-beam) and nondegenerate (pump- and signal-beam) cases are considered.

© 2002 Optical Society of America

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    [CrossRef]
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2002 (1)

A. R. Cowan and J. F. Young, “Mode matching for second harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

2001 (2)

V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602 (2001).
[CrossRef]

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, “Resonant scattering and mode coupling in two-dimensional textured planar waveguides,” J. Opt. Soc. Am. A 18, 1160–1171 (2001).
[CrossRef]

2000 (6)

A. Hache and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089–4091 (2000).
[CrossRef]

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

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

E. Garmire, “Resonant optical nonlinearities in semiconductors,” IEEE J. Sel. Top. Quantum Electron. 6, 1094–1110 (2000).
[CrossRef]

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

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

1999 (3)

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

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

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

1997 (1)

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

1996 (1)

D. M. Atkin, P. St. J. Russell, T. A. Birks, and 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]

1995 (2)

1994 (1)

1992 (1)

1991 (1)

1988 (1)

1985 (2)

Assanto, G.

Astratov, V. N.

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

Atkin, D. M.

D. M. Atkin, P. St. J. Russell, T. A. Birks, and 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, and 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]

Bourgeois, M.

A. Hache and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089–4091 (2000).
[CrossRef]

Busch, A.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

Centeno, E.

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

Cowan, A. R.

A. R. Cowan and J. F. Young, “Mode matching for second harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, “Resonant scattering and mode coupling in two-dimensional textured planar waveguides,” J. Opt. Soc. Am. A 18, 1160–1171 (2001).
[CrossRef]

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

Culshaw, I. S.

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

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and 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, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60, R16255–R16258 (1999).
[CrossRef]

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

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

Felbacq, D.

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

Fortenberry, R. M.

Garmire, E.

E. Garmire, “Resonant optical nonlinearities in semiconductors,” IEEE J. Sel. Top. Quantum Electron. 6, 1094–1110 (2000).
[CrossRef]

Hache, A.

A. Hache and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089–4091 (2000).
[CrossRef]

Hagan, D. J.

Joannopoulos, J. D.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

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

Johnson, S. G.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

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

Johnson, S. R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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, and 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, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Kolodziejski, L. A.

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

Koster, A.

Krauss, T. F.

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

Lee, R. K.

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

Liao, C.

Lousse, V.

V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602 (2001).
[CrossRef]

Mackenzie, J.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

Mandeville, J.

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

Marques, M. B.

Morin, R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

Moshrefzadeh, R.

Nevière, M.

O’Brien, J. D.

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

Pacradouni, V.

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, “Resonant scattering and mode coupling in two-dimensional textured planar waveguides,” J. Opt. Soc. Am. A 18, 1160–1171 (2001).
[CrossRef]

V. Pacradouni, J. Mandeville, A. R. Cowan, P. Paddon, and 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, and 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]

Paddon, P.

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, “Resonant scattering and mode coupling in two-dimensional textured planar waveguides,” J. Opt. Soc. Am. A 18, 1160–1171 (2001).
[CrossRef]

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

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

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

Painter, O.

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

Paraire, N.

Popov, E.

Reinisch, R.

Roberts, P. J.

D. M. Atkin, P. St. J. Russell, T. A. Birks, and 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, and 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]

Said, A. A.

Scherer, A.

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

Seaton, C. T.

Sheik-Bahae, M.

Shoemaker, R. L.

Skolnick, M. S.

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

Stegeman, G. I.

Stevenson, R. M.

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

Tiedje, T.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

Tran, P.

P. Tran, “Photonic band-structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B 52, 10673–10676 (1995).
[CrossRef]

Van Stryland, E. W.

Velera, J. D.

Vigneron, J. P.

V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602 (2001).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

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

Vincent, P.

Wang, J.

Wei, T. H.

Whittaker, D. M.

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

Yariv, A.

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

Young, J.

Young, J. F.

A. R. Cowan and J. F. Young, “Mode matching for second harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, “Resonant scattering and mode coupling in two-dimensional textured planar waveguides,” J. Opt. Soc. Am. A 18, 1160–1171 (2001).
[CrossRef]

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

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

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

Appl. Phys. Lett. (2)

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and 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]

A. Hache and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089–4091 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. Garmire, “Resonant optical nonlinearities in semiconductors,” IEEE J. Sel. Top. Quantum Electron. 6, 1094–1110 (2000).
[CrossRef]

J. Mod. Opt. (1)

D. M. Atkin, P. St. J. Russell, T. A. Birks, and 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 (3)

J. Opt. Soc. Am. B (5)

Phys. Rev. B (8)

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

P. Paddon and 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. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and 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, and J. F. Young, “Photonic bandstructure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

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

P. Tran, “Photonic band-structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B 52, 10673–10676 (1995).
[CrossRef]

A. R. Cowan and J. F. Young, “Mode matching for second harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

Phys. Rev. E (1)

V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602 (2001).
[CrossRef]

Science (1)

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

Other (2)

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

S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Institute of Physics Publishing, Bristol, UK, 1995).

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Figures (11)

Fig. 1
Fig. 1

Schematic representation of a 2-D waveguide-based photonic crystal. Here we consider a square lattice of cylindrical holes on pitch Λ with radius r that completely penetrate the GaAs slab of thickness w. This porous core rests upon a dielectric cladding layer of thickness d that separates it from a GaAs substrate. Plane waves of arbitrary polarization are incident with in-plane wave vectors β .

Fig. 2
Fig. 2

Linear specular reflectivity for s-polarized (solid curve) and p-polarized (dashed curve) incident plane waves with an in-plane wave vector of β = 0.01 β g oriented along the Γ X direction of the square lattice.

Fig. 3
Fig. 3

Dispersion of the lowest-lying four modes near zone center, as identified in Fig. 2. Each band has a different line type that corresponds to those used in Fig. 5, below. These bands are labeled in order of increasing energy as s1, s2, p, and s3.

Fig. 4
Fig. 4

Real space plots of the magnitude of the total electric field within the unit cell for (a) s1, (b) s2, (c) p, and (d) s3, all corresponding to the modes that are apparent in Fig. 2. Each circle represents the perimeter of the air hole etched in the core layer. The horizontal and the vertical scales are in units of nanometers, and the gray scales are arbitrary in an absolute sense but consistent from plot to plot.

Fig. 5
Fig. 5

Inverse Q (FWHM divided by the center frequency) of the same bands shown in Fig. 3 (with corresponding line types in the two figures).

Fig. 6
Fig. 6

Degenerate reflectivity in the vicinity of the s3 mode in Fig. 2 for various incident field intensities: linear limit, dashed curve. Progressively blueshifted solid curves show intensities of 765, 2342, and 3871 kW/cm2 from left to right.

Fig. 7
Fig. 7

Degenerate reflectivity in the vicinity of the s3 mode in Fig. 2. The linear limit is shown by a dashed curve. The solid curve shows the nonlinear result when the off-diagonal elements of the χ ( 3 ) tensor are equal to the diagonal elements, and the dashed–dotted curve results when the off-diagonal elements are set to zero. The intensity in this case is 4779 kW/cm2.

Fig. 8
Fig. 8

Degenerate reflectivity in the vicinity of the s1 mode in Fig. 2 for various incident field intensities: linear limit, dashed curve. Progressively blueshifted solid curves show intensities of 30, 48, 107, and 155 kW/cm2 from left to right.

Fig. 9
Fig. 9

Bistable behavior (degenerate) of the reflectivity in the vicinity of the s3 resonance in Fig. 2.

Fig. 10
Fig. 10

Nondegenerate response of the s1 mode to various pump intensities with the pump frequency fixed at 10 744 cm-1. β is fixed at 0.01 β g x ˆ for all pump and probe frequencies. The intensities of the pump beams are 1720 (solid curve) and 3058 (dashed–dotted curve) kW/cm2. The dashed curve represents the linear (low-intensity limit) spectrum.

Fig. 11
Fig. 11

Nondegenerate response of the s3 mode to different pump intensities with the pump frequency fixed at 10 744 cm-1. β is fixed at 0.01 β g x ˆ for all pump and probe frequencies. The intensities of the pump beams are 1720 (solid curve) and 3058 (dashed–dotted curve) kW/cm2. The dashed curve represents the linear (low-intensity limit) spectrum.

Equations (17)

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E ( ω : β inc + β n ) = E hom ( ω : β inc + β n ) + G ( ω : β inc + β n ) Δ P ( ω : β inc + β n ) ,
Δ P ( ω : β inc + β n ) = Δ χ ( 1 ) ( - ω ; ω : β n - β m ) × E ( ω : β inc + β m )
Δ P ( ω : β inc + β n )
= Δ χ ( 1 ) ( - ω ; ω : β n - β k ) E ( ω : β inc + β k ) + χ ( 3 ) ( - ω ; ω , - ω ,   ω : β n - β m + β l - β k ) × E ( ω : β inc + β m ) E * ( ω : - β inc - β l ) E ( ω : β inc + β k ) ,
Δ P μ = Δ χ μ γ ( 1 ) E γ + χ μ α δ γ ( 3 ) E α E δ * E γ .
Δ P μ = [ Δ χ μ γ ( 1 ) + θ μ γ ( 3 ) ] E γ = χ μ γ eff E γ ,
θ μ γ ( 3 ) ( ω : β n - β k )
= χ μ α δ γ ( 3 ) ( - ω ; ω , - ω ,   ω : β n - β m + β l - β k ) × E α ( ω : β inc + β m ) E δ * ( - β inc - β l ) ,
χ μ γ eff = Δ χ μ γ ( 1 ) + θ μ γ ( 3 ) = Δ χ μ γ ( 1 ) + χ μ α δ γ ( 3 ) E α E δ * ,
E α ( ω : β inc + β n ) = E α hom ( ω : β inc + β n ) + G α μ ( ω : β inc + β n ) × χ μ γ eff ( β n - β k ) E γ ( ω : β inc + β k ) .
χ xxxx ( 3 ) = χ yyyy ( 3 ) = χ zzzz ( 3 ) ,
χ xxyy ( 3 ) = χ xxzz ( 3 ) = χ yyxx ( 3 ) = χ yyzz ( 3 ) = χ zzxx ( 3 ) = χ zzyy ( 3 ) ,
χ xyxy ( 3 ) = χ xzxz ( 3 ) = χ yxyx ( 3 ) = χ yzyz ( 3 ) = χ zyzy ( 3 ) = χ zxzx ( 3 ) ,
χ xyyx ( 3 ) = χ xzzx ( 3 ) = χ yxxy ( 3 ) = χ yzzy ( 3 ) = χ zyyz ( 3 ) = χ zxxz ( 3 ) ,
E α ( ω s ,   β probe + β n )
= E α hom ( ω s ,   β probe + β n ) + G α μ ( ω s ,   β probe + β n ) × [ Δ χ μ γ ( 1 ) ( - ω s ; ω s : β n - β k ) E γ ( ω s : β probe + β k ) + χ μ α δ γ ( 3 ) ( - ω s ; ω p , - ω p ,   ω s : β n - β m + β l - β k ) × E α ( ω p : β pump + β m ) E δ * ( ω p : - β pump - β l ) × E γ ( ω s : β probe + β k ) ] .
χ μ γ eff = Δ χ μ γ ( 1 ) ( - ω s ; ω s ) + χ μ α δ γ ( 3 ) × ( - ω s ; ω p , - ω p ,   ω s ) E α ( ω p ) E δ * ( ω p ) .

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