Abstract

We describe stable symmetry-breaking states in systems with two coupled nonlinear cavities, using coupled-mode theory and rigorous simulations. Above a threshold input level the symmetric state of the passive Kerr system becomes unstable, and we show how this phenomenon can be employed for switching and flip-flop purposes, using positive pulses only. A device with compact photonic crystal microcavities is proposed by which we numerically demonstrate the principle.

© 2006 Optical Society of America

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  1. T. Yabuzaki, T. Okamoto, M. Kitano, and T. Ogawa, "Optical bistability with symmetry-breaking," Phys. Rev. A 29,1964-1972 (1984).
    [CrossRef]
  2. K. Otsuka and K. Ikeda, "Hierarchical multistability and cooperative flip-flop operation in a bistable optical system with distributed nonlinear elements," Opt. Lett. 12,599-601 (1987).
    [CrossRef] [PubMed]
  3. K. Otsuka, "All-optical flip-flop operations in a coupled element bistable device," Electron. Lett. 24,800-801 (1988).
    [CrossRef]
  4. K. Otsuka, "Pitchfork bifurcation and all-optical digital signal-processing with a coupled-element bistable system," Opt. Lett. 14,72-74 (1989).
    [CrossRef] [PubMed]
  5. M. Haelterman and P. Mandel, "Pitchfork bifurcation using a 2-beam nonlinear fabry-perot-interferometer-an analytical study," Opt. Lett. 15,1412-1414 (1990).
    [CrossRef] [PubMed]
  6. T. Peschel, U. Peschel, and F. Lederer, "Bistability and symmetry-breaking in distributed coupling of counterpropagating beams into nonlinear wave-guides," Phys. Rev. A 50,5153-5163 (1994).
    [CrossRef] [PubMed]
  7. I. V. Babushkin, Y. A. Logvin, and N. A. Lo?ko, "Symmetry breaking in optical dynamics of two bistable thin films," Quantum Electron. 28,104-107 (1998).
    [CrossRef]
  8. J. P. Torres, J. Boyce, and R. Y. Chiao, "Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability," Phys. Rev. Lett. 83,4293-4296 (1999).
    [CrossRef]
  9. L. Longchambon, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, "Experimental evidence of spontaneous symmetry breaking in intracavity type II second-harmonic generation with triple resonance," Opt. Lett. 30,284-286 (2005).
    [CrossRef] [PubMed]
  10. M. Solja?i?, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R) (2002).
    [CrossRef]
  11. 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]
  12. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13,2678-2687 (2005).
    [CrossRef] [PubMed]
  13. P. E. Barclay, K. Srinivasan, and O. Painter, "Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper," Opt. Express 13,801-820 (2005).
    [CrossRef] [PubMed]
  14. B. Maes, P. Bienstman, and R. Baets, "Switching in coupled nonlinear photonic-crystal resonators," J. Opt. Soc. Am. B 22,1778-1784 (2005).
    [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. M. F. Yanik, S. Fan, and M. Solja?i?, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83,2739-2741 (2003).
    [CrossRef]
  18. P. V. Paulau and N. A. Lo?ko, "Self-sustained pulsations of light in a nonlinear thin-film system," Phys. Rev. A 72,013819 (2005).
    [CrossRef]
  19. M. Solja?i? and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature Materials 3,211-219 (2004).
    [CrossRef] [PubMed]
  20. Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30,1989-1991 (2005).
    [CrossRef] [PubMed]

2005 (6)

2004 (2)

M. Solja?i? and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature Materials 3,211-219 (2004).
[CrossRef] [PubMed]

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 (1)

M. F. Yanik, S. Fan, and M. Solja?i?, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83,2739-2741 (2003).
[CrossRef]

2002 (1)

2001 (1)

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]

1999 (1)

J. P. Torres, J. Boyce, and R. Y. Chiao, "Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability," Phys. Rev. Lett. 83,4293-4296 (1999).
[CrossRef]

1998 (1)

I. V. Babushkin, Y. A. Logvin, and N. A. Lo?ko, "Symmetry breaking in optical dynamics of two bistable thin films," Quantum Electron. 28,104-107 (1998).
[CrossRef]

1994 (1)

T. Peschel, U. Peschel, and F. Lederer, "Bistability and symmetry-breaking in distributed coupling of counterpropagating beams into nonlinear wave-guides," Phys. Rev. A 50,5153-5163 (1994).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

1988 (1)

K. Otsuka, "All-optical flip-flop operations in a coupled element bistable device," Electron. Lett. 24,800-801 (1988).
[CrossRef]

1987 (1)

1984 (1)

T. Yabuzaki, T. Okamoto, M. Kitano, and T. Ogawa, "Optical bistability with symmetry-breaking," Phys. Rev. A 29,1964-1972 (1984).
[CrossRef]

Babushkin, I. V.

I. V. Babushkin, Y. A. Logvin, and N. A. Lo?ko, "Symmetry breaking in optical dynamics of two bistable thin films," Quantum Electron. 28,104-107 (1998).
[CrossRef]

Baets, R.

B. Maes, P. Bienstman, and R. Baets, "Switching in coupled nonlinear photonic-crystal resonators," J. Opt. Soc. Am. B 22,1778-1784 (2005).
[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]

Barclay, P. E.

Bienstman, P.

B. Maes, P. Bienstman, and R. Baets, "Switching in coupled nonlinear photonic-crystal resonators," J. Opt. Soc. Am. B 22,1778-1784 (2005).
[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]

Boyce, J.

J. P. Torres, J. Boyce, and R. Y. Chiao, "Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability," Phys. Rev. Lett. 83,4293-4296 (1999).
[CrossRef]

Chiao, R. Y.

J. P. Torres, J. Boyce, and R. Y. Chiao, "Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability," Phys. Rev. Lett. 83,4293-4296 (1999).
[CrossRef]

Coudreau, T.

Fabre, C.

Fan, S.

Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30,1989-1991 (2005).
[CrossRef] [PubMed]

M. F. Yanik, S. Fan, and M. Solja?i?, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83,2739-2741 (2003).
[CrossRef]

Haelterman, M.

Ikeda, K.

Joannopoulos, J. D.

M. Solja?i? and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature Materials 3,211-219 (2004).
[CrossRef] [PubMed]

Kira, G.

Kitano, M.

T. Yabuzaki, T. Okamoto, M. Kitano, and T. Ogawa, "Optical bistability with symmetry-breaking," Phys. Rev. A 29,1964-1972 (1984).
[CrossRef]

Kivshar, Y. S.

Kuramochi, E.

Laurat, J.

Lederer, F.

T. Peschel, U. Peschel, and F. Lederer, "Bistability and symmetry-breaking in distributed coupling of counterpropagating beams into nonlinear wave-guides," Phys. Rev. A 50,5153-5163 (1994).
[CrossRef] [PubMed]

Logvin, Y. A.

I. V. Babushkin, Y. A. Logvin, and N. A. Lo?ko, "Symmetry breaking in optical dynamics of two bistable thin films," Quantum Electron. 28,104-107 (1998).
[CrossRef]

Loiko, N. A.

P. V. Paulau and N. A. Lo?ko, "Self-sustained pulsations of light in a nonlinear thin-film system," Phys. Rev. A 72,013819 (2005).
[CrossRef]

I. V. Babushkin, Y. A. Logvin, and N. A. Lo?ko, "Symmetry breaking in optical dynamics of two bistable thin films," Quantum Electron. 28,104-107 (1998).
[CrossRef]

Longchambon, L.

Maes, B.

B. Maes, P. Bienstman, and R. Baets, "Switching in coupled nonlinear photonic-crystal resonators," J. Opt. Soc. Am. B 22,1778-1784 (2005).
[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]

Mandel, P.

Mingaleev, S. F.

Mitsugi, S.

Notomi, M.

Ogawa, T.

T. Yabuzaki, T. Okamoto, M. Kitano, and T. Ogawa, "Optical bistability with symmetry-breaking," Phys. Rev. A 29,1964-1972 (1984).
[CrossRef]

Okamoto, T.

T. Yabuzaki, T. Okamoto, M. Kitano, and T. Ogawa, "Optical bistability with symmetry-breaking," Phys. Rev. A 29,1964-1972 (1984).
[CrossRef]

Otsuka, K.

Painter, O.

Paulau, P. V.

P. V. Paulau and N. A. Lo?ko, "Self-sustained pulsations of light in a nonlinear thin-film system," Phys. Rev. A 72,013819 (2005).
[CrossRef]

Peschel, T.

T. Peschel, U. Peschel, and F. Lederer, "Bistability and symmetry-breaking in distributed coupling of counterpropagating beams into nonlinear wave-guides," Phys. Rev. A 50,5153-5163 (1994).
[CrossRef] [PubMed]

Peschel, U.

T. Peschel, U. Peschel, and F. Lederer, "Bistability and symmetry-breaking in distributed coupling of counterpropagating beams into nonlinear wave-guides," Phys. Rev. A 50,5153-5163 (1994).
[CrossRef] [PubMed]

Shinya, A.

Soljacic, M.

M. Solja?i? and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature Materials 3,211-219 (2004).
[CrossRef] [PubMed]

M. F. Yanik, S. Fan, and M. Solja?i?, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83,2739-2741 (2003).
[CrossRef]

Srinivasan, K.

Tanabe, T.

Torres, J. P.

J. P. Torres, J. Boyce, and R. Y. Chiao, "Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability," Phys. Rev. Lett. 83,4293-4296 (1999).
[CrossRef]

Treps, N.

Wang, Z.

Yabuzaki, T.

T. Yabuzaki, T. Okamoto, M. Kitano, and T. Ogawa, "Optical bistability with symmetry-breaking," Phys. Rev. A 29,1964-1972 (1984).
[CrossRef]

Yanik, M. F.

M. F. Yanik, S. Fan, and M. Solja?i?, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83,2739-2741 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

M. F. Yanik, S. Fan, and M. Solja?i?, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83,2739-2741 (2003).
[CrossRef]

Electron. Lett. (1)

K. Otsuka, "All-optical flip-flop operations in a coupled element bistable device," Electron. Lett. 24,800-801 (1988).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nature Materials (1)

M. Solja?i? and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature Materials 3,211-219 (2004).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (5)

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. A (3)

P. V. Paulau and N. A. Lo?ko, "Self-sustained pulsations of light in a nonlinear thin-film system," Phys. Rev. A 72,013819 (2005).
[CrossRef]

T. Yabuzaki, T. Okamoto, M. Kitano, and T. Ogawa, "Optical bistability with symmetry-breaking," Phys. Rev. A 29,1964-1972 (1984).
[CrossRef]

T. Peschel, U. Peschel, and F. Lederer, "Bistability and symmetry-breaking in distributed coupling of counterpropagating beams into nonlinear wave-guides," Phys. Rev. A 50,5153-5163 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. P. Torres, J. Boyce, and R. Y. Chiao, "Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability," Phys. Rev. Lett. 83,4293-4296 (1999).
[CrossRef]

Quantum Electron. (1)

I. V. Babushkin, Y. A. Logvin, and N. A. Lo?ko, "Symmetry breaking in optical dynamics of two bistable thin films," Quantum Electron. 28,104-107 (1998).
[CrossRef]

Other (1)

M. Solja?i?, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic of the coupled cavity structure. (b) The PhC device, with 4 periods in between the switches. An electric field distribution is superimposed to illustrate the defect modes.

Fig. 2.
Fig. 2.

Output power versus input power for (a) ∆ = 1.039, ϕ = 0.570 and (b) ∆ = 2.0, ϕ = 0.595. Stable and unstable states are shown with solid and dashed lines, respectively. Dots show rigorous simulation results.

Fig. 3.
Fig. 3.

Output powers versus left input power PinL at ∆ = 1.039, ϕ = 0.570 and PinR = 3.125P 0. Stable states for PoutR (resp. PoutL ) are shown with red (resp. blue) solid lines. Dashed lines indicate unstable states. Dots (resp. circles) show rigorous simulation results for PoutR (resp. PoutL ). Labels AB, CD and EF display key states.

Fig. 4.
Fig. 4.

Switching of the state by adding power to the left input PinL Here ∆ = 1.039, ϕ = 0.570 and the period is 2π/ω. After an initial small perturbation of the right input power it is held constant at PinR = 3.125P 0. The dotted (dashed) line shows PinL (PinR ). The blue and red lines depict PoutL and PoutR , respectively.

Equations (6)

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

d a j d t = [ i ( ω 0 + δ ω j ) 1 τ ] a j + d f j + d b j + 1 ,
b j = exp ( i ϕ ) f j + d a j ,
f j + 1 = exp ( i ϕ ) b j + 1 + d a j ,
[ i ( ω 0 ω + δ ω 1 ) 1 τ ] a 1 + κ ( γ a 1 + a 2 ) = d f 1 ,
[ i ( ω 0 ω + δ ω 2 ) 1 τ ] a 2 + κ ( γ a 2 + a 1 ) = d b 3 ,
( A B ) [ ( A 2 + A B + B 2 ) + 2 Δ ( A + B ) + Δ 2 + 1 4 ] = 0 ,

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