Abstract

Numerical simulations have shown the existence of transversely localized guided modes in nonlinear two-dimensional photonic crystals. These soliton-like Bloch waves induce their own waveguide in a photonic crystal without the presence of a linear defect. By applying a Green’s function method which is limited to within a strip perpendicular to the propagation direction, we are able to describe these Bloch modes by a nonlinear lattice model that includes the long-range site-to-site interaction between the scattered fields and the non-local nonlinear response of the photonic crystal. The advantages of this semi-analytical approach are discussed and a comparison with a rigorous numerical analysis is given in different configurations. Both monoatomic and diatomic nonlinear photonic crystals are considered.

© 2005 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).
  2. H. M. Gibbs, Optical bistability: controlling light with light (Academic press, Orlando, 1985).
  3. V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602 (pages 4) (2001).
    [Crossref]
  4. P. Tran, “Photonic band structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B 52, 10,673–10,676 (1995).
    [Crossref]
  5. P. Tran, “Optical switching with a nonlinear photonic crystal: a numerical study,” Opt. Lett. 21, 1138–1134 (1995).
    [Crossref]
  6. A. J. Sievers and S. Takeno, “Intrinsic localized modes in anharmonic crystals,” Phys. Rev. Lett. 61, 970–973 (1988).
    [Crossref] [PubMed]
  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. D. Cai, A. Bishop, and N. Gronbech-Jensen, “Localized states in discrete nonlinear schrodinger equations,” Phys. Rev. Lett. 72, 591–595 (1994).
    [Crossref] [PubMed]
  9. 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]
  10. 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]
  11. 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]
  12. F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial solitons, chap. Discrete solitons, 269–292 (Springer, Berlin, 2001).
  13. 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]
  14. B. Maes, P. Bienstman, and R. Baets, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, ? (2005). To be published.
    [Crossref]
  15. 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]
  16. B. Maes, P. Bienstman, and R. Baets, “Modeling of Kerr nonlinear photonic components with mode expansion,” Opt. Quantum. Electron. 36, 15–24 (2004).
    [Crossref]
  17. S. F. Mingaleev and Y. S. Kivshar, “Effective equations for photonic-crystal waveguides and circuits,” Opt. Lett. 27, 231–233 (2002).
    [Crossref]
  18. S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19, 2241–2249 (2002).
    [Crossref]
  19. H. Ammari and F. Santosa, “Guided waves in a photonic bandgap structure with a line defect,” SIAM J. Appl. Math. 64, 2018–2033 (2004).
    [Crossref]
  20. C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4mm photonic crystals,” Phys. Rev. B 56, 7313–7320 (1997).
    [Crossref]
  21. S. F. Mingaleev and Y. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
    [Crossref] [PubMed]

2005 (1)

B. Maes, P. Bienstman, and R. Baets, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, ? (2005). To be published.
[Crossref]

2004 (2)

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

H. Ammari and F. Santosa, “Guided waves in a photonic bandgap structure with a line defect,” SIAM J. Appl. Math. 64, 2018–2033 (2004).
[Crossref]

2002 (3)

2001 (3)

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 (pages 4) (2001).
[Crossref]

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

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).
[Crossref]

1999 (1)

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]

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).
[Crossref]

1997 (1)

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4mm photonic crystals,” Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

1995 (2)

P. Tran, “Photonic band structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B 52, 10,673–10,676 (1995).
[Crossref]

P. Tran, “Optical switching with a nonlinear photonic crystal: a numerical study,” Opt. Lett. 21, 1138–1134 (1995).
[Crossref]

1994 (1)

D. Cai, A. Bishop, and N. Gronbech-Jensen, “Localized states in discrete nonlinear schrodinger equations,” Phys. Rev. Lett. 72, 591–595 (1994).
[Crossref] [PubMed]

1993 (1)

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

1988 (1)

A. J. Sievers and S. Takeno, “Intrinsic localized modes in anharmonic crystals,” Phys. Rev. Lett. 61, 970–973 (1988).
[Crossref] [PubMed]

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]

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

Ammari, H.

H. Ammari and F. Santosa, “Guided waves in a photonic bandgap structure with a line defect,” SIAM J. Appl. Math. 64, 2018–2033 (2004).
[Crossref]

Anderson, C. M.

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4mm photonic crystals,” Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

Baets, R.

B. Maes, P. Bienstman, and R. Baets, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, ? (2005). To be published.
[Crossref]

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, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, ? (2005). To be published.
[Crossref]

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]

Bishop, A.

D. Cai, A. Bishop, and N. Gronbech-Jensen, “Localized states in discrete nonlinear schrodinger equations,” Phys. Rev. Lett. 72, 591–595 (1994).
[Crossref] [PubMed]

Cai, D.

D. Cai, A. Bishop, and N. Gronbech-Jensen, “Localized states in discrete nonlinear schrodinger equations,” Phys. Rev. Lett. 72, 591–595 (1994).
[Crossref] [PubMed]

Darmanyan, S.

F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial solitons, chap. Discrete solitons, 269–292 (Springer, Berlin, 2001).

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]

Giapis, K. P.

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4mm photonic crystals,” Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

Gibbs, H. M.

H. M. Gibbs, Optical bistability: controlling light with light (Academic press, Orlando, 1985).

Gronbech-Jensen, N.

D. Cai, A. Bishop, and N. Gronbech-Jensen, “Localized states in discrete nonlinear schrodinger equations,” Phys. Rev. Lett. 72, 591–595 (1994).
[Crossref] [PubMed]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

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]

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.

Kobyakov, A.

F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial solitons, chap. Discrete solitons, 269–292 (Springer, Berlin, 2001).

Lederer, F.

F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial solitons, chap. Discrete solitons, 269–292 (Springer, Berlin, 2001).

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 (pages 4) (2001).
[Crossref]

Maes, B.

B. Maes, P. Bienstman, and R. Baets, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, ? (2005). To be published.
[Crossref]

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

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

Mingaleev, S. F.

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

S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19, 2241–2249 (2002).
[Crossref]

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

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]

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]

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]

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]

Santosa, F.

H. Ammari and F. Santosa, “Guided waves in a photonic bandgap structure with a line defect,” SIAM J. Appl. Math. 64, 2018–2033 (2004).
[Crossref]

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]

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]

Tran, P.

P. Tran, “Photonic band structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B 52, 10,673–10,676 (1995).
[Crossref]

P. Tran, “Optical switching with a nonlinear photonic crystal: a numerical study,” Opt. Lett. 21, 1138–1134 (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 (pages 4) (2001).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Opt. Quantum. Electron. (2)

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]

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

Phys. Rev. B (2)

P. Tran, “Photonic band structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B 52, 10,673–10,676 (1995).
[Crossref]

C. M. Anderson and K. P. Giapis, “Symmetry reduction in group 4mm photonic crystals,” Phys. Rev. B 56, 7313–7320 (1997).
[Crossref]

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 (pages 4) (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. (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]

A. J. Sievers and S. Takeno, “Intrinsic localized modes in anharmonic crystals,” Phys. Rev. Lett. 61, 970–973 (1988).
[Crossref] [PubMed]

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

D. Cai, A. Bishop, and N. Gronbech-Jensen, “Localized states in discrete nonlinear schrodinger equations,” Phys. Rev. Lett. 72, 591–595 (1994).
[Crossref] [PubMed]

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

SIAM J. Appl. Math. (1)

H. Ammari and F. Santosa, “Guided waves in a photonic bandgap structure with a line defect,” SIAM J. Appl. Math. 64, 2018–2033 (2004).
[Crossref]

Other (3)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

H. M. Gibbs, Optical bistability: controlling light with light (Academic press, Orlando, 1985).

F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial solitons, chap. Discrete solitons, 269–292 (Springer, Berlin, 2001).

Supplementary Material (2)

» Media 1: GIF (1230 KB)     
» Media 2: GIF (1010 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

The band-gap structure of the photonic crystal consisting of a square lattice of square dielectric rods (n=3.4) in an air background. The ratio of the side of the rods to the crystal period is d/a=0.25.

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Fig. 2.
Fig. 2.

A gap soliton with ω=0.38(2πc/a) and kx =0.7π/a propagating along the x-direction. [Media 1]

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Fig. 3.
Fig. 3.

(a) Photonic crystal geometry with Green’s function at ω=0.38(2πc/a) superimposed. (b) Strip Green’s function at ω k x = 0.38 ( 2 π c ⁄ a ) and kx =0.7π/a calculated using Eq. (6). In both cases r 2=0.

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Fig. 4.
Fig. 4.

Coupling parameters Jn and electric field at the center of rod n for a self-localized waveguide with ω k x and kx =0.7(π/a).

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Fig. 5.
Fig. 5.

Modal energy Q and modal electric field amplitudes of the self-localized waveguide at ω k x = 0.38 ( 2 π c ⁄ a ) , calculated with strip’s Green theory (thin line) and exact simulations (thick line), respectively. The field in the center of rod 0 (black), 1 (red), 2 (green) and 3 (blue) are shown.

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Fig. 6.
Fig. 6.

(a) Diatomic photonic crystal geometry with Green’s function at ω=0.4(2πc/a) superimposed. (b) Strip Green’s function at ω k x = 0.4 ( 2 π c ⁄ a ) and kx =0.85π/a.

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Fig. 7.
Fig. 7.

Modal energy Q for the diatomic photonic crystal at ω k x = 0.4 ( 2 π c ⁄ a ) , calculated with strip’s Green theory (thin line) and exact simulations (thick line), respectively.

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Fig. 8.
Fig. 8.

A propagating diatomic gap soliton with ω k x = 0.4 ( 2 π c ⁄ a ) and kx =0.908π/a. [Media 2]

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Equations (13)

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

[ ∇ 2 + ε ( r ) ( ω c ) 2 ] E ( r | ω ) = 0 .
ε L ( r + r a x + s a y ) = ε L ( r ) ,
[ ∇ 2 + ε L ( r ) ( ω k x c ) 2 ] E ( r | ω k x , k x ) = 0 .
[ ∇ 2 + ( ω k x c ) 2 ε L ( x ) ] g ( r 1 , r 2 ∣ ω k x , k x ) = ∑ j ∈ ℤ δ ( r 1 + j a x − r 2 ) e i j k x a ,
g ( r 1 + a x , r 2 | ω k x , k x ) = g ( r 1 , r 2 | ω k x , k x ) e i k x a
g ( r 1 , r 2 | ω k x , k x ) = ∑ j ∈ ℤ G ( r 1 + j a x , r 2 | ω ) e i j k x a .
ε ( r ) = ε L ( r ) + ε N L ( r ) .
[ ∇ 2 + ( ω k x c ) 2 ε L ( r ) ] E ( r | ω k x , k x ) = − ( ω k x c ) 2 ε N L ( r ) E ( r | ω k x , k x ) .
E ( r | ω k x , k x ) = ( ω k x c ) 2 ∫ strip g ( r , u | ω k x , k x ) ε N L ( u ) E ( u | ω k x , k x ) d 2 u .
ε N L ( r ) = − δ rod ( r ) ∣ E ( r ∣ ω k x , k x ) ∣ 2 ,
E n ( ω k x , k x ) = − ∑ m J n , m ( ω k x | k x ) ∣ E m ( ω k x , k x ) ∣ 2 E m ( ω k x , k x ) ,
J l ( ω k x , k x ) = ( ω k x c ) 2 ∫ rod g ( r 0 , r 1 + u | ω k x , k x ) d 2 u ,
Q = ∑ m ∣ E m ( ω k x , k x ) ∣ 2 .

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