Abstract

We describe the effect of symmetry breaking in a system with two coupled Fano resonators. A general criterion is derived and optimal parameter regions for switching are identified. By extending the single resonator behavior we show that one achieves the effect at very low input powers.

©2008 Optical Society of America

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References

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  • Publication
  1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Physical Review 124, 1866 (1961).
    [Crossref]
  2. V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. E 69, 155106 (2004).
  3. J.E. Heebner, R.W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
    [Crossref]
  4. B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005).
    [Crossref]
  5. B. Maes, M. Soljačić, J.D. Joannopoulos, P. Bienstman, R. Baets, S-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10678.
    [Crossref] [PubMed]
  6. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-801.
    [Crossref] [PubMed]
  7. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2678.
    [Crossref] [PubMed]
  8. T. Uesugi, B-S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377.
    [Crossref] [PubMed]
  9. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,” Opt. Lett. 30, 3069–3071 (2005).
    [Crossref] [PubMed]
  10. S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002).
    [Crossref]
  11. 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]
  12. S. Fan, W. Suh, and J.D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
    [Crossref]
  13. M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

2006 (2)

2005 (4)

2004 (1)

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

2003 (1)

2002 (2)

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]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Physical Review 124, 1866 (1961).
[Crossref]

Asano, T.

Baets, R.

Barclay, P.E.

Bienstman, P.

Boyd, R.W.

Fan, S.

S. Fan, W. Suh, and J.D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002).
[Crossref]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Physical Review 124, 1866 (1961).
[Crossref]

Fink, Y.

M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

Gorza, S-P.

Haelterman, M.

Heebner, J.E.

Ibanescu, M.

M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

Joannopoulos, J.D.

Johnson, S.G.

M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

Kira, G.

Kuramochi, E.

Li, X.

Lousse, V.

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

Lu, Y.

Maes, B.

Mitsugi, S.

Noda, S.

Notomi, M.

Painter, O.

Park, Q-H.

Shinya, A.

Soljacic, M.

Song, B-S.

Srinivasan, K.

Suh, W.

Tanabe, T.

Uesugi, T.

Vigneron, J.P.

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

Wang, P.

Yao, J.

Appl. Phys. Lett. (1)

S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002).
[Crossref]

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

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

Opt. Express (4)

Opt. Lett. (1)

Opt. Quantum Electron. (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]

Phys. Rev. E (1)

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

Physical Review (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Physical Review 124, 1866 (1961).
[Crossref]

Other (1)

M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

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Figures (5)
Fig. 1.
Fig. 1. (a) Schematic of the two-cavity device (bottom) and electric field plot of a device with three blocking rods per cavity (top). (b) Transmission T (black line) and reflection amplitude r = 1 − T (red line) versus number of blocking rods in the PhC waveguide at wavelength a/0.367.

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Fig. 2.
Fig. 2. Transmission T in three different cavity structures: (a) no blocking rods (Tbackground ≈1); (b) one blocking rod with side 0.2a (Tbackground ≈0.5); (c) three blocking rods (Tbackground ≈0).

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Fig. 3.
Fig. 3. (a) Critical detunings Δ± for r=1 (black lines), r=0.8 (blue lines) and r=0 (red lines), respectively. The largest value is Δ+, the smallest corresponds to Δ-. (b) Minimum input power Pmin in and corresponding Pmax in versus ϕ for r=1.0 (black lines) and r=0.8 (red lines), respectively. For these results Pmin in was minimized. All powers in units of P 0.

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Fig. 4.
Fig. 4. (a) Asymmetric states for r=1, ϕ=0.5π with Δ=1.02 (black dots) and Δ=0.62 (red dots), respectively. The symmetric states are shown with a line. (b) Linear transmission T versus detuning Δ with r=1 for ϕ=0.5π (ϕ=0.1π) in black (red), respectively.

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Fig. 5.
Fig. 5. Asymmetric states for (a) r=0.8, ϕ=0.6π, Δ=1.25 and (b) r=0.8, ϕ=0.23π, Δ=2.1. The symmetric states are shown with a line. (c) Linear transmission T versus detuning Δ with r=0.8 for ϕ=0.6π (ϕ=0.23π) in black (red), respectively.

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Equations (8)

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d a j d t = [ i ( ω 0 + δ ω j ) − 1 τ ] a j + df j + db j + 1 ,
[ b j f j + 1 ] = exp ( i ϕ ) [ r it it r ] · [ f j b j + 1 ] + d a j [ 1 1 ] ,
[ i ( ω 0 − ω + δ ω 1 ) − 1 τ ] a 1 + κd ( c 1 a 1 + a 2 ) = − ( d + κc 1 c 2 ) f 1 − κc 2 b 3 ,
[ i ( ω 0 − ω + δ ω 2 ) − 1 τ ] a 2 + κd ( c 1 a 2 + a 1 ) = − ( d + κc 1 c 2 ) b 3 − κc 2 f 1 ,
( A − B ) [ ( A 2 + AB + B 2 ) + 2 ( Δ + Γ 1 ) ( A + B ) + Δ 2 + 2 Δ Γ 1 + Γ 2 ] = 0 ,
Γ 1 = ( rt + r sin ⁡ ϕ + t cos ⁡ ϕ ) ( 1 + 2 r cos ⁡ ϕ + r 2 ) ,
Γ 2 = ( 1 + 2 t sin ⁡ ϕ + t 2 ) ( 1 + 2 r cos ⁡ ϕ + r 2 ) .
Δ ± = − Γ 1 ± 3 ( Γ 2 − Γ 1 2 ) .

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