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).
    [CrossRef] [PubMed]
  2. D. Cai, A. R. Bishop, and N. Gronbech-Jensen, "Localized states in discrete nonlinear Schrödinger equations," Phys. Rev. Lett. 72, 591-595 (1994).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  9. B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  13. P. Tran, "Photonic-band-structure calculation of material possessing Kerr nonlinearity," Phys. Rev. B 52, 10673-10676 (1995).
    [CrossRef]
  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).
    [CrossRef]
  15. A. Huttunen and P. Törmä, "Band structures for nonlinear photonic crystals," J. Appl. Phys. 91, 3988-3991 (2002).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  19. R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
    [CrossRef]

2004

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

2003

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).
[CrossRef] [PubMed]

2002

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

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).
[CrossRef]

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).
[CrossRef]

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]

2000

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).
[CrossRef]

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).
[CrossRef]

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]

1998

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).
[CrossRef]

1995

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

1994

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

1993

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

1991

1988

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

1964

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

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).
[CrossRef]

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).
[CrossRef]

Aközbek, N.

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

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Chiao, R. Y.

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

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Garmire, E.

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

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Huttunen , A.

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

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).
[CrossRef] [PubMed]

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).
[CrossRef]

John , S.

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

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef]

Scherer, A.

Sievers , A. J.

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

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Törmä, P.

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

Townes, C. H.

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

Tran, P.

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

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]

Xu, Y.

Yariv, A.

J. Appl. Phys.

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

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

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

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).
[CrossRef]

Phys. Rev. B

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

Phys. Rev. E

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]

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).
[CrossRef]

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).
[CrossRef]

Phys. Rev. Lett.

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

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).
[CrossRef]

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

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

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).
[CrossRef] [PubMed]

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

Other

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.

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/.

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Figures (8)

Fig. 1
Fig. 1

Simple structure with three invariant sections along z.

Fig. 2
Fig. 2

Example of spatial discretization. The middle section is considered nonlinear.

Fig. 3
Fig. 3

Nonlinear photonic crystal. (a) Geometry: The rectangular box shows the unit cell that was used, exploiting symmetry. (b) Band structure: The solid curves show linear bands and dashed curves present nonlinearly adjusted bands.

Fig. 4
Fig. 4

Coupled-cavity waveguide. (a) Geometry: The rectangular box shows the unit cell used. Period p=3a. (b) Band structure: curves, linear bands with defect rod index n_cen; filled circles and crosses, nonlinearly adjusted bands.

Fig. 5
Fig. 5

Photonic crystal limiter. (a) Geometry: defect waveguide with larger nonlinear rods alongside. The rectangular box shows the unit cell used. (b) Waveguide bands: solid curves, linear bands; dashed curves, nonlinearly adjusted bands. The upper bands overlap. Period p=2a.

Fig. 6
Fig. 6

Examples of the electric fields of two types of staggered self-localized waveguide. The smaller rectangular boxes show the unit cell that we used to find the mode. The larger rectangular (dashed) boxes are used for the study of stability. The mode is linearly propagated through a crystal with a refractive-index distribution derived from the nonlinear calculation, which is limited to the small rectangular box. Insets, electric-field profiles in a transversal cut through the rod centers. (a) On-site mode with a/λ=0.38 and P=1.2×10-10 W/m. (b) Intersite mode with a/λ=0.41 and P=9×10-10 W/m.

Fig. 7
Fig. 7

Energy Ue versus power P for self-localized waveguides of different wavelengths. For values of a/λ, o and i mean on-site and intersite, respectively.

Fig. 8
Fig. 8

Width σ versus maximum refractive-index change Δnmax for on-site modes of various wavelengths a/λ. Insets, electric-field profiles for the three indicated points.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

E(r)=iAiEi(rt)exp(-jβiz),
H(r)=iAiHi(rt)exp(-jβiz),
n(r)=n0+n2I(r),
F2=T12F1+R21B2,
B1=R12F1+T21B2,
T12R210IF1qB1=qI0R12T21F1qB1,
P=12 (E×H*)·dS,
Ue=02 n2IdV.
ω=ω0+Δω cos(kzp),
σ=i di2Eimax/j Ejmax1/2,

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