Abstract

We introduce simple model equations describing the dynamics of light in thin photonic crystal films with Kerr nonlinearity. We report modulational instabilities and bright and dark localized structures of light that exist in this system in the proximity of Fano resonances.

© 2005 Optical Society of America

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References

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  1. N.N. Rosanov, “Transverse patterns in wide-aperture nonlinear optical systems,” Prog. in Opt. 35, 1–60 (1996).
    [Crossref]
  2. W.J. Firth and G.K. Harkness , “Existence, stability, and properties of cavity solitons,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 343–358.
  3. S. Trillo and M. Haeleterman , “Parametric solitons in passive structures with feedback,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 359–394.
  4. S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
    [Crossref] [PubMed]
  5. B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
    [Crossref] [PubMed]
  6. K. Staliunas, “Midband dissipative spatial solitons,” Phys. Rev. Lett. 91, 053901–053905 (2003).
    [Crossref] [PubMed]
  7. U. Peschel, O. Egorov, and F. Lederer, “Discrete cavity soliton,” Opt. Lett. 29, 1909–1911 (2004).
    [Crossref] [PubMed]
  8. D. Gomila, R. Zambrini, and G.-L. Oppo, “Photonic band-gap inhibition of modulational instabilities,” Phys. Rev. Lett. 92, 253904–253908 (2004).
    [Crossref] [PubMed]
  9. J.M. Pottage, E. Silvestre, and P.St.J. Russell, “Vertical-cavity surface-emitting resonances in photonic crystal films,” J. Opt. Soc. Am. A 18, 442–447 (2001).
    [Crossref]
  10. S.H. Fan and J.D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112–235120 (2002).
    [Crossref]
  11. V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. B 69, 155106–155117 (2004).
    [Crossref]
  12. A. R. Cowan and J.F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606–046622 (2003).
    [Crossref]
  13. N. Akzbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
    [Crossref]
  14. C.M. de Sterke, B.J. Eggleton, and J.E. Sipe , “Bragg solitons: Theory and experiment,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 169–210.
  15. K. Ogusu, J. Yamasaki, and Sh. Maeda, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett.,  29, 265–267 (2004).
    [Crossref] [PubMed]

2004 (4)

U. Peschel, O. Egorov, and F. Lederer, “Discrete cavity soliton,” Opt. Lett. 29, 1909–1911 (2004).
[Crossref] [PubMed]

D. Gomila, R. Zambrini, and G.-L. Oppo, “Photonic band-gap inhibition of modulational instabilities,” Phys. Rev. Lett. 92, 253904–253908 (2004).
[Crossref] [PubMed]

V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. B 69, 155106–155117 (2004).
[Crossref]

K. Ogusu, J. Yamasaki, and Sh. Maeda, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett.,  29, 265–267 (2004).
[Crossref] [PubMed]

2003 (2)

A. R. Cowan and J.F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606–046622 (2003).
[Crossref]

K. Staliunas, “Midband dissipative spatial solitons,” Phys. Rev. Lett. 91, 053901–053905 (2003).
[Crossref] [PubMed]

2002 (2)

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

S.H. Fan and J.D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112–235120 (2002).
[Crossref]

2001 (1)

2000 (1)

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref] [PubMed]

1998 (1)

N. Akzbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[Crossref]

1996 (1)

N.N. Rosanov, “Transverse patterns in wide-aperture nonlinear optical systems,” Prog. in Opt. 35, 1–60 (1996).
[Crossref]

Ackemann, T.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref] [PubMed]

Akzbek, N.

N. Akzbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[Crossref]

Balle, S.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Barland, S.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Brambilla, M.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Cowan, A. R.

A. R. Cowan and J.F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606–046622 (2003).
[Crossref]

de Sterke, C.M.

C.M. de Sterke, B.J. Eggleton, and J.E. Sipe , “Bragg solitons: Theory and experiment,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 169–210.

Eggleton, B.J.

C.M. de Sterke, B.J. Eggleton, and J.E. Sipe , “Bragg solitons: Theory and experiment,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 169–210.

Egorov, O.

Fan, S.H.

S.H. Fan and J.D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112–235120 (2002).
[Crossref]

Feldmann, M.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref] [PubMed]

Firth, W.J.

W.J. Firth and G.K. Harkness , “Existence, stability, and properties of cavity solitons,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 343–358.

Giudici, M.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Gomila, D.

D. Gomila, R. Zambrini, and G.-L. Oppo, “Photonic band-gap inhibition of modulational instabilities,” Phys. Rev. Lett. 92, 253904–253908 (2004).
[Crossref] [PubMed]

Haeleterman, M.

S. Trillo and M. Haeleterman , “Parametric solitons in passive structures with feedback,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 359–394.

Harkness, G.K.

W.J. Firth and G.K. Harkness , “Existence, stability, and properties of cavity solitons,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 343–358.

Jager, R.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Joannopoulos, J.D.

S.H. Fan and J.D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112–235120 (2002).
[Crossref]

John, S.

N. Akzbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[Crossref]

Knodl, T.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Lange, W.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref] [PubMed]

Lederer, F.

Lousse, V.

V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. B 69, 155106–155117 (2004).
[Crossref]

Lugiato, L.A.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Maeda, Sh.

Maggipinto, T.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Miller, M.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Ogusu, K.

Oppo, G.-L.

D. Gomila, R. Zambrini, and G.-L. Oppo, “Photonic band-gap inhibition of modulational instabilities,” Phys. Rev. Lett. 92, 253904–253908 (2004).
[Crossref] [PubMed]

Peschel, U.

Pottage, J.M.

Rosanov, N.N.

N.N. Rosanov, “Transverse patterns in wide-aperture nonlinear optical systems,” Prog. in Opt. 35, 1–60 (1996).
[Crossref]

Russell, P.St.J.

Schäpers, B.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref] [PubMed]

Silvestre, E.

Sipe, J.E.

C.M. de Sterke, B.J. Eggleton, and J.E. Sipe , “Bragg solitons: Theory and experiment,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 169–210.

Spinelli, L.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Staliunas, K.

K. Staliunas, “Midband dissipative spatial solitons,” Phys. Rev. Lett. 91, 053901–053905 (2003).
[Crossref] [PubMed]

Tissoni, G.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Tredicce, J.R.

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Trillo, S.

S. Trillo and M. Haeleterman , “Parametric solitons in passive structures with feedback,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 359–394.

Vigneron, J.P.

V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. B 69, 155106–155117 (2004).
[Crossref]

Yamasaki, J.

Young, J.F.

A. R. Cowan and J.F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606–046622 (2003).
[Crossref]

Zambrini, R.

D. Gomila, R. Zambrini, and G.-L. Oppo, “Photonic band-gap inhibition of modulational instabilities,” Phys. Rev. Lett. 92, 253904–253908 (2004).
[Crossref] [PubMed]

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

Nature (1)

S. Barland, J.R. Tredicce, M. Brambilla, L.A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (2)

S.H. Fan and J.D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112–235120 (2002).
[Crossref]

V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. B 69, 155106–155117 (2004).
[Crossref]

Phys. Rev. E (2)

A. R. Cowan and J.F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606–046622 (2003).
[Crossref]

N. Akzbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[Crossref]

Phys. Rev. Lett. (3)

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref] [PubMed]

K. Staliunas, “Midband dissipative spatial solitons,” Phys. Rev. Lett. 91, 053901–053905 (2003).
[Crossref] [PubMed]

D. Gomila, R. Zambrini, and G.-L. Oppo, “Photonic band-gap inhibition of modulational instabilities,” Phys. Rev. Lett. 92, 253904–253908 (2004).
[Crossref] [PubMed]

Prog. in Opt. (1)

N.N. Rosanov, “Transverse patterns in wide-aperture nonlinear optical systems,” Prog. in Opt. 35, 1–60 (1996).
[Crossref]

Other (3)

W.J. Firth and G.K. Harkness , “Existence, stability, and properties of cavity solitons,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 343–358.

S. Trillo and M. Haeleterman , “Parametric solitons in passive structures with feedback,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 359–394.

C.M. de Sterke, B.J. Eggleton, and J.E. Sipe , “Bragg solitons: Theory and experiment,” in ‘Spatial Solitons’, Eds. S. Trillo and W. Torruelas (Springer, 2001), pp. 169–210.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic view of the system. The top grating provides Fano resonance and the bottom grating provides Bragg scattering between the guided resonant modes. The guided modes have the wave vectors ±β, k 0 is the wave vector and θ is the incident angle of the pump wave.

Fig. 2.
Fig. 2.

(a),(b) Dependence of the normalized energy density Jnorm2·(|A +|2+|A -|2)/I 2 on the frequency detuning δ for q=-1 and Γ=7.55·10-4. The dashed line corresponds to the linear case with pump I=5.5·10-6 and the solid line corresponds to the nonlinear case with pump I=5.5·10-5. Bistability appears in the nonlinear case, see full lines. Modulational instability on the positive slope can appear also; this region is situated between points A and B in (b). (c) and (d) show the behavior of the reflection coefficients on the pump frequency detuning for the linear (dashed line) and nonlinear (solid line) cases.

Fig. 3.
Fig. 3.

The same as Fig. 2, but for q=0.

Fig. 4.
Fig. 4.

The dependence of the energy density W=|A +|2+|A -|2 of the spatially uniform solution and on the maximum of W=|A +|2+|A -|2 of the bright LS on the pump amplitude I. Frequency detuning δ=0.9978, wave vector detuning q=0 and dissipation Γ=7.55·10-4. The upper branch of the spatially uniform state is modulationally unstable.

Fig. 5.
Fig. 5.

The dependences of the energy density W=|A +|2+|A -|2 in bright (a) and dark (b) solitons are shown for the case with I=4.4·10-5, Γ=7.55·10-4. For bright solitons the pump frequency is δ=0.9977 and for dark solitons the frequency is δ=-1.00238.

Equations (2)

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( i t + i x + i Γ + A + 2 + 2 A 2 ) A + + A = I e iqx i δ t
( i t i x + i Γ + A 2 + 2 A + 2 ) A + A + = 0 .

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