Abstract

We present a modeling technique that uses eigenmode expansion to simulate infinite periodic structures with Kerr nonlinearity. Using a unit cell with Bloch boundary conditions, our iterative algorithm efficiently calculates self-consistent two-dimensional Bloch modes. We show how it can be used to study the band structure of nonlinear photonic crystals and to gain rapid insight in the operation of devices. Furthermore, we present nonlinear transversely localized guided modes, which are kinds of gap solitons or intrinsic localized modes, that induce their own waveguide through a photonic crystal without linear defects.

© 2005 Optical Society of America

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  1. A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).
  2. D. Cai, A. R. Bishop, and N. Gronbech-Jensen, "Localized states in discrete nonlinear Schrödinger equations," Phys. Rev. Lett. 72, 591-595 (1994).
  3. S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).
  4. A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear guided waves and spatial solitons in a periodic layered medium," J. Opt. Soc. Am. B 19, 772-781 (2002).
  5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
  6. F. Lederer, S. Darmanyan, and A. Kobyakov, "Discrete solitons," in Spatial Solitons , S. Trillo and W. Torruellas, eds. (Springer-Verlag, Berlin, 2001), pp. 269-292.
  7. S. John and N. Aközbek, "Nonlinear optical solitary waves in a photonic band gap," Phys. Rev. Lett. 71, 1168-1171 (1993).
  8. N. Aközbek and S. John, "Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures," Phys. Rev. E 57, 2287-2319 (1998).
  9. B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
  10. P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
  11. CAMFR simulation software is freely available from http://camfr.sourceforge.net/.
  12. K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).
  13. P. Tran, "Photonic-band-structure calculation of material possessing Kerr nonlinearity," Phys. Rev. B 52, 10673-10676 (1995).
  14. V. Lousse and J. P. Vigneron, "Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials," Phys. Rev. E 63, 027602 (2001).
  15. A. Huttunen and P. Törmä, "Band structures for nonlinear photonic crystals," J. Appl. Phys. 91, 3988-3991 (2002).
  16. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
  17. T. Fujisawa and M. Koshiba, "Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits," in Integrated Photonics Research , OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2003), p. IME5.
  18. A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, "Self-induced optical fibers - spatial solitary waves," Opt. Lett. 16, 21-23 (1991).
  19. R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).

2004 (1)

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).

2003 (1)

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).

2002 (2)

A. Huttunen and P. Törmä, "Band structures for nonlinear photonic crystals," J. Appl. Phys. 91, 3988-3991 (2002).

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear guided waves and spatial solitons in a periodic layered medium," J. Opt. Soc. Am. B 19, 772-781 (2002).

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).

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).

2000 (1)

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).

1999 (2)

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

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).

1998 (1)

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

1995 (1)

P. Tran, "Photonic-band-structure calculation of material possessing Kerr nonlinearity," Phys. Rev. B 52, 10673-10676 (1995).

1994 (1)

D. Cai, A. R. Bishop, and N. Gronbech-Jensen, "Localized states in discrete nonlinear Schrödinger equations," Phys. Rev. Lett. 72, 591-595 (1994).

1993 (1)

S. John and N. Aközbek, "Nonlinear optical solitary waves in a photonic band gap," Phys. Rev. Lett. 71, 1168-1171 (1993).

1991 (1)

1988 (1)

A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).

Aközbek , N.

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

Aközbek, N.

S. John and N. Aközbek, "Nonlinear optical solitary waves in a photonic band gap," Phys. Rev. Lett. 71, 1168-1171 (1993).

Baets, R.

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).

Bienstman, P.

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).

Bienstman , P.

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).

Bishop, A. R.

D. Cai, A. R. Bishop, and N. Gronbech-Jensen, "Localized states in discrete nonlinear Schrödinger equations," Phys. Rev. Lett. 72, 591-595 (1994).

Cai, D.

D. Cai, A. R. Bishop, and N. Gronbech-Jensen, "Localized states in discrete nonlinear Schrödinger equations," Phys. Rev. Lett. 72, 591-595 (1994).

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).

Fan, S.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).

Gronbech-Jensen, N.

D. Cai, A. R. Bishop, and N. Gronbech-Jensen, "Localized states in discrete nonlinear Schrödinger equations," Phys. Rev. Lett. 72, 591-595 (1994).

Huang, K. C.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).

Huttunen , A.

A. Huttunen and P. Törmä, "Band structures for nonlinear photonic crystals," J. Appl. Phys. 91, 3988-3991 (2002).

Joannopoulos, J. D.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).

John, S.

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

John , S.

S. John and N. Aközbek, "Nonlinear optical solitary waves in a photonic band gap," Phys. Rev. Lett. 71, 1168-1171 (1993).

Kivshar, Y. S.

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear guided waves and spatial solitons in a periodic layered medium," J. Opt. Soc. Am. B 19, 772-781 (2002).

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).

Ladouceur, F.

Lee, R. K.

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).

Maes, B.

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).

Mingaleev, S. F.

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).

Mitchell, D. J.

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).

Nelson, K. A.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).

Poladian, L.

Sammut, R. A.

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).

Scherer, A.

Sievers , A. J.

A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).

Silberberg, Y.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).

Snyder, A. W.

Sukhorukov , A. A.

Takeno, S.

A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).

Törmä, P.

A. Huttunen and P. Törmä, "Band structures for nonlinear photonic crystals," J. Appl. Phys. 91, 3988-3991 (2002).

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).

Tran, P.

P. Tran, "Photonic-band-structure calculation of material possessing Kerr nonlinearity," Phys. Rev. B 52, 10673-10676 (1995).

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).

Xu, Y.

Yariv, A.

J. Appl. Phys. (1)

A. Huttunen and P. Törmä, "Band structures for nonlinear photonic crystals," J. Appl. Phys. 91, 3988-3991 (2002).

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

Opt. Lett. (2)

Opt. Quantum Electron. (2)

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).

Phys. Rev. B (1)

P. Tran, "Photonic-band-structure calculation of material possessing Kerr nonlinearity," Phys. Rev. B 52, 10673-10676 (1995).

Phys. Rev. E (3)

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

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

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).

Phys. Rev. Lett. (6)

A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).

D. Cai, A. R. Bishop, and N. Gronbech-Jensen, "Localized states in discrete nonlinear Schrödinger equations," Phys. Rev. Lett. 72, 591-595 (1994).

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, "Field expulsion and reconfiguration in polaritonic photonic crystals," Phys. Rev. Lett. 90, 196402 (2003).

R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).

S. John and N. Aközbek, "Nonlinear optical solitary waves in a photonic band gap," Phys. Rev. Lett. 71, 1168-1171 (1993).

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).

Other (3)

F. Lederer, S. Darmanyan, and A. Kobyakov, "Discrete solitons," in Spatial Solitons , S. Trillo and W. Torruellas, eds. (Springer-Verlag, Berlin, 2001), pp. 269-292.

CAMFR simulation software is freely available from http://camfr.sourceforge.net/.

T. Fujisawa and M. Koshiba, "Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits," in Integrated Photonics Research , OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2003), p. IME5.

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