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

Full Article  |  PDF Article

References

  • View by:
  • |

  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 onlinearity,�?? 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]

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 onlinearity,�?? 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)

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

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).

Supplementary Material (2)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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.

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]

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.

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).

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.

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.

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.

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]

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 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 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 .

Metrics